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Hamlet. 
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Henry  1 
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Henry  1 
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Richard 
Richard 
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Romeo 
Othello. 
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IN  MEMOmAM 
John  3v/ett 


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^ell. 


Vol. 


:  aim 
text 
and 


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Habpebs'  Graded  Abitiwetj'cs 


SECOND  BOOK  IN  ARITHMETIC 


COMPRISING 

FOUR  YEARS  OF  ORAL  AND  WRITTEN  WORK 
IN  THE  ELEMENTS  OF  NUMBERS 


NEAV    YORK 
HARPER  &  BROTHERS,  FRANKLIN  SQUARE 

1883 


^A.f 


Copyright,  1882,  by  Harper  &  Brothers. 

All  rights  reserved. 


PREFACE. 


This  is  the  second  of  a  series  of  two  books  embracing  a  com- 
plete course  in  arithmetic  both  oral  and  written,  for  schools  be- 
low the  high-school  grade. 

The  first  276  pages  comprise  a  brief  course  in  the  rudiments 
of  arithmetic,  inckiding,  in  their  order,  integers,  decimals,  prop- 
erties of  numbers,  fractions,  compound  numbers,  measurements, 
and  percentage;  and  closing  with  outlines  for  reviews  and  exami- 
nations. 

The  subjects  of  this  brief  course  are  those  essential  to  prepa- 
ration for  business,  or  for  advanced  study;  and  they  are  as  fully 
treated  as  the  average  age  and  advancement  of  pupils  in  such  a 
course  will  justif}^ 

Oral  and  written  work  are  combined  throughout  the  brief 
course.  Every  method  of  computation  is  introduced  by-  simple 
problems  that  contain  only  small  numbers,  and  can  be  solved 
orally.  These  are  followed  by  similar  problems  that  contain 
larger  numbers,  and  require  written  solutions. 

The  arrangement  of  the  matter  conforms  to  the  synthetic  or 
inductive  method  of  instruction ;  every  principle  and  process  of 
computation,  and  every  essential  fact  or  principle  being  devel- 
oped by  inductive  questions. 

The  definitions,  principles,  and  rules  are  simple,  clear,  concise, 
and  comprehensive.  All  rules  for  computations  are  based  upon 
principles  previously  deduced  by  inductive  questioning ;  and  the 
number  of  definitions,  principles,  and  rules  is   reduced  to  tbe 


mmimura. 


4  PREFACE, 

Incomplete  forms  of  the  elementary  combinations  are  given 
in  tables,  which  tlie  pupil  is  required  to  copy,  complete,  and 
memorize. 

The  oral  exercises  in  adding,  subtracting,  multiplying,  and  di- 
viding in  series,  and  the  written  exercises  in  forming  combina- 
tions at  sight,  secure  accuracy  and  rapidity  in  the  use  of  numbers. 

The  problems  in  written  work  are  ample  for  all  needed  drill 
and  practice.  They  furnish  copious  illustrations  of  methods  and 
processes,  and  abundant  exercise  in  the  application  of  these  meth- 
ods and  processes  to  business  computations  in  the  various  occupa- 
tions, trades,  and  professions. 

Pages  277-364  comprise  matter  strictly  supplementary  to  the 
subjects  in  the  brief  course.  They  contain  articles  on  methods 
of  proof,  general  principles  of  division  and  fractions,  short  meth- 
ods of  computations,  converse  reductions;  price,  quantity,  and 
cost ;  longitude  and  time ;  compound  numbers,  the  metric  sys- 
tem, percentage,  proportion,  involution,  and  evolution ;  measure- 
ments, and  forms  of  business  paper. 

The  arrangement  of  this  supplement  is  such,  that  the  study  of 
the  several  subjects  may  immediately  follow  the  study  of  the 
same  subjects  in  the  body  of  the  book ;  or  the  supplement  may 
be  taken  up  in  course,  after  the  study  of  the  body  of  the  book 
is  completed. 

The  book  contains  as  much  work  in  the  theory  and  practice 
of  numbers  as  can  profitably  be  done  in  four  school  years  of 
eight  to  ten  months  each,  —  the  brief  course  occupying  three 
years,  and  the  supplement  one  year. 

Two  editions  of  this  book  are  published — one  with  answers  to 
the  problems,  and  the  other  without  answers. 

The  work  is  submitted  to  the  public,  with  the  confident  belief 
that  it  contains  all  the  essentials  of  a  model  text-book. 


CONTENTS. 


Integers 7-110 


Notation  and  numeration  .     .      7 

Addition 17 

Subtraction ........     83 


Multiplication 50 

Division 74 

Sixteen  general  problems  .     .  101 


Decimals 111-140 


Notation  and  numeration  .     .111 

Reduction 119 

Addition 121 


Subtraction 123 

Multiplication 126 

Division 180 


Properties  of  Numbers 


141-152 


Factors  and  divisors. 


.  141 1  Multiples 146 

Cancellation 150 


Fractions 153-188 


Notation  and  numeration  .     .153 

Reductions 156 

Addition 166 


Subtraction 169 

Multiplication 172 

Division 179 


Compound  Numbers 189-210 

Measures 189  I  Time .  204 

Weights 202  1  Enumeration  or  counting  .     .  207 


Measurements 


Rectangles,  triangles,  trapezoids  211 

The  circle 219 

Rectangular  solids 222 


Business  measurements 
Measures  of  surface  . 
Measures  of  volume     . 


211-236 

.  .  225 
.  .  225 
.     .  226 


Measures  of  farm  products 230 


CONTENTS. 


Percentage .    237-268 


Notation 237 

The  five  general  cases  .     .     .  239 

Profit  and  loss 246 

Commission 247 

Insurance 249 


Taxes 250 

Customs  or  duties     ....  251 

Stocks 252 

Partnership 254 

Interest 257 


Blackboard  Outlines 269-276 


SUPPLEIVIENT 277-364 


Methods  of  proof  .  .  .  .277 
Principles  of  division  .  .  279 
Principles  of  fractions  .     .     .  280 

Short  methods 281 

Converse  reductions ....  284 
Price,  quantity,  and  cost  .  .  288 
Compound  numbers ....  293 
Longitude  and  time  ....  299 
Metric  measures  and  weights .  302 

Interest .  306 

Business  papers 314 


Discount 315 

Compound  interest    ....  320 

Partial  payments 322 

Bonds 327 

Equation  of  payments  .     .     .  329 

Exchange 330 

Ratio  and  proportion     .     .     .  333 

Powers  and  roots 342 

Measurements 352 

Forms  of  business  papers  .     .  360 

Tax,  interest,  and  coin  tables.  363 


SECOND  BOOK  IN  ARITHMETIC. 


CHAPTER    I. 

INTEGERS. 


SECTION   I. 

NOTATION  AND  NUMERATION. 

1 .  Arithmetic  is  tlie  science  of  numbers  and  the  art 
of  using  them. 

S.  A  tiiiit  is  a  single  thing  or  one. 
One  apple,  one  dollar,  one  dozen,  one  hundred,  are  units. 

3.  A  miniber  is  a  unit,  or  two  or  more  units. 

One,  fifteen,  forty-six  apples,  two  hundred  pounds,  are  numbers. 

4.  The  iiiiit  of  a  ntirriber  is  a  single  thing  or  one, 

of  the  kind  expressed  by  the  number. 

The  unit  of  fifteen  is  one ;  of  forty  apples,  one  apple ;  of  two  hun- 
dred twenty-five  pounds,  one  pound. 

5.  An  integer  is  a  number  that  expresses  whole  or 
undivided  things. 

Integers  are  also  called  tvJiole  nunibet^s, 

6.  Ten  figures  are  used  in  writing  numbers.    They  are 

0123456789 

a.  The  naught  is  often  called  cipher  or  zero, 
h.  The  other  figures  are  called  digits. 


8  SECOJUD   BOOK  IN  ARITHMETIC. 

7.  To  express  an  integer  not  greater  than  nine,  only 

oiiu  figure  is  used, 

8.  To  express  an  integer  greater  than  nine,  two  or 
more  figures  are  used. 


3B&&&3&&03 


{{{(au{{4 


Ten  ones 


one  ten. 


9.  Ten  ones  taken  together  are  a  ten. 
Ten  is  written  10 


Two  tens  are  written  20 
Three  tens "  "  30 
Four  tens  "  "  40 
Five  tens    "         "       50 


Six  tens  are  written  60 

Seven  tens  are    "  70 

Eight  tens    "     "  80 

Nine  tens     "     "  90 


10,  When  an  integer  is  expressed  by  two  figures,  the 
left-hand  figure  expresses  the  tens,  and  the  right-hand  fig- 
ure the  ones. 

1  ten  and  6  ones  are  written  16 
3  tens  and  5  ones,  written  35     17  tens  and  0  ones,  written  70 
5  tens    "     8    "  "       58     I    9  tens    "     1  one,         "       91 


//////////'i 


Ten  tens 


are      one  hundred. 


11.  Ten  tens  taken  together  are  a  hundred. 
One  hundred  is  written  100 


Two  hundred  is  written  200 
Three  hundred  '*  "  300 
Four  hundred  "  "  400 
Five  hundred    "       "       500 


Six  hundred  is  written  600 
Seven  hundred  "  "  700 
Eight  hundred  "  "  800 
Nine  hundred   "       "       900 


INTEGERS.— NOTATION  AND   NUMERATION      9 


1 S.  When  an  integer  is  expressed  by  three  figures,  the 
left-hand  figure  expresses  hundreds ;  the  middle  figure, 
tens ;  and  the  right-hand  figure,  ones, 

2  hundreds,  4  tens,  3  ones  are  two  hundred  forty-three,  written  243 
4  hundreds,  6  tens,  2  ones  are  four  hundred  sixty-two,  "  462 
6  hundreds,  7  tens,  0  ones  are  six  hundred  seventy,  "       670 

8  hundred?,  0  tens,  4  ones  are  eight  hundred  four,  "       804 

9  hundreds,  1  ten,  2  ones  are  nine  hundred  twelve,  "      912 

EXEECISES. 

A.  Write  and  read  the  numbers  expressed  by 

6  hundreds  5  tens  and  1  one. 

"    0  ones. 

"    8     " 
u    3     u 

"    5     " 


1  hundred  2  tens  and  4  ones. 

2  hundreds  2  "  "  4  " 
o  «  3  "  "  4  *' 
4  "  4  "  «  4  " 
6        "        3    "      "    7     " 


JB.  Eead 
12         64 
61  99 

80  76 

28  17 


O.  Write  in  words 


130 
216 
952 
305 


427 
745 
810 
685 


13 
71 

20 
59 


94 

47 


611 
193 
308 
572 


821 
450 
734 
206 


Z>.  Express  by  figures 


i.  Eighty-two.  Jf.  Forty. 

^.  Twenty-seven.  5.  Sixty-nine. 

3.  Ninety-three.  6.  Fifty-five. 

10.  One  hundred  seventeen. 

11.  Two  hundred  ninety-one. 

12.  Four  hundred  sixty-three. 

13.  Six  hundred  seventy-six. 
IJf..  Three  hundred  thirty-eight. 

15.  Seven  hundred  fifty-two. 

16.  Eight  hundred  twenty-four. 

17.  Nine  hundred  forty-five. 

18.  Five  hundred  eighty-nine. 

A  2 


7,  Seventy-nine. 

8,  Fourteen. 

9,  Thirty-seven. 

19.  Nine  hundred. 

20.  Five  hundred  sixty. 

21.  Three  hundred  ten. 

22.  Six  hundred  ninety. 

23.  Eight  hundred  forty. 
2Jf..  Six  hundred  six. 

25.  Seven  hundred  two. 

26.  Five  hundred  seven. 

27.  Four  hundred  nine. 


10 


SECOND   BOOK  IN  ARITHMETIC. 


lillliil 


Ten  hundreds 


are       one  thousand. 


13.  Ten  hundreds  taken  together  are  a  thousand. 
One  thousand  is  written  1,000 


Two  thousand,  2,000 
Three  thousand,  3,000 
Four  thousand,  4,000 
Five  thousand,     5,000 


Six  thousand,  6,000 
Seven  thousand,  7,000 
Eight  thousand,  8,000 
Nine  thousand,    9,000 


14.  Ten  thousands  taken  together  are  a  ten-thousand. 
Ten  ten-thousands  taken  together  are  a  hundred-thou- 
sand. 

One  ten- thousand  is  written  10,000 

Two  ten-thousands  are  twenty  thousand,  written  20,000 
Eight  ten-thousands  are  eighty  thousand,  "  80,000 
One  hundred-thousand  is  written  100,000 

Five  hundred-thousands  are  written  500,000 

15.  When  an  integer  is  expressed  by  more  than  three 
figures,  the  figure  at  the  left  of  hundreds  expresses  thou- 
sands', the  figure  at  the  left  of  thousands  expresses  ten- 
thousands ;  and  the  figure  at  the  left  of  ten  -  thousands 
expresses  hundred-thousands. 

1 6.  Every  three  figures  in  an  integer,  8  7  4,  235 
counting  from  the  right,  are  a  period*  |  *  •  i  •  • 
Periods  of  figures  are  separated  by  com-  I  *  s  s  S  S 
mas.  A  ^  %     *  5  o 

The  first  or  right-hand  period  consists  of  ones^  tens,  and 


INTEGERS.— NOTATION  AND   NUMERATION     11 


hundreds ;  and  the  second  period  of  07ies  of  tliousands, 
tens  of  thousands,  and  hundreds  of  thousands. 

Eighteen  thousand  five  hundred  thirty-six  is  written    18,536 


Thirty-two  thousand  eight 
Forty-seven  thousand  two  hundred  " 

Sixty  thousand  four  hundred  twenty  " 

Two  hundred  forty  thousand  " 

Four  hundred  eight  thousand  five  hundred  " 
Five  hundred  thousand  three  hundred  two  " 
Six  hundred  fifty-two  thousand  ten  " 

Eight  hundred  fifty  thousand  sixty-one       " 

EXEECISES. 


809,051 

40,269 

660,020 


A.  Eead 

8,000 

26,506 

574,000 

5,400 

37,081 

629,005 

2,560 

93,274 

700,044 

J5.  Express  by  words 


6,014 
3,405 
4,009 


33,000 
80,900 
90,209 


19,040 

855,480 
300,070 


530,240 

902,105 

20,007 


32,008 
47,200 
60,420 
240,000 
408,500 
500,302 
652,010 
850,061 


100,905 

50,041 

482,070 


404,040 

40,040 

876,543 


C.  Express  by  figures 

1.  Five  thousand ;  sixty  thousand ;  two  hundred  thou- 
sand ;  seven  hundred  sixty-five  thousand. 

^.  Nine  thousand  eight  hundred;  two  thousand  six; 
seventy-two  thousand  four  hundred ;  seven  hundred  forty- 
seven  thousand  two  hundred. 

3.  Eifty  thousand  twenty ;  seven  hundred  ten  thou- 
sand ;  eight  thousand  fifty. 

Jf..  Eight  hundred  one  thousand  four ;  two  hundred 
thousand  six  hundred  forty ;  four  thousand  fifty-six. 

5.  Sixty-three  thousand  four;  six  thousand  eight  hun- 
dred nineteen ;  three  hundred  thirty  thousand  seventy. 


12  SECOND    BOOK  IN  ARITHMETIC, 

6.  One  hundred  five  thousand  four  hundred  seventy; 
seven  thousand  two  hundred  forty ;  nineteen  thousand 
three  hundred  one ;  fifty-six  thousand  eleven. 

7.  Five  hundred  forty  thousand  seventy-two ;  one  thou- 
sand nine  hundred  three ;  nine  hundred  fifty-seven  thou- 
sand five  hundred  three. 

8.  Nine  hundred  thousand  six  hundred ;  six  hundred 
eighty  thousand  five ;  three  hundred  sixty-five  thousand 
two  hundred  ninety-four. 

9.  One  thousand  ten;  fifty  thousand  three  hundred 
sixty-seven;  four  hundred  sixty  thousand  twenty;  eight 
hundred  three  thousand. 

10.  Five  hundred  thousand  three  hundred  eighty-four ; 
twelve  thousand  nineteen ;  nine  hundred  nine  thousand 
nine  hundred  nine. 

1 7,  The  third  period  of  figures  expresses  ones  of  mill- 
ions, tens  of  millions,  and  hundreds  of  millions ;  and  the 
fourth  period  expresses  ones  of  billions,  tens  of  billions, 
and  hundreds  of  billions. 

Two  million  five  thousand  eighty  is  written       2,005,080 

Thirty-four  billion  three  hundred  twenty- 
four  thousand  five  hundred  eighty-six     "      "    34,000,324,586 

Forty  million  forty-four  thousand  twelve    "      "  40,044,012 

One  hundred  twenty-nine  million  three  hun- 
dred seventeen  thousand-five  hundred     "      "         129,317,500 

Seven  hundred  fifty  million  two  hundred 

thousand  seventy  "      "         750,200,070 

Nine  hundred  three  billion  fifty  million 

five  hundred  ninety-four  thousand  "      "  903,050,594,000 

Four  billion   six  hundred  million   seven 

hundred  eighty-six  "        "      4,600,000,786 

18.  In  any  period  the  right-hand  figure  expresses  ones, 
the  second  figure  expresses  tens,  and  the  third  figure  ex- 
presses hundreds. 


INTEGERS.—NOTATION  AND   NUMERATION.     13 


19.  The  order  of  a  unit  takes  its  name  from  the 
place  it  occupies. 

A  digit  written  in  the  first  place,  expresses  units  of  the  first  order; 
in  the  second  place,  it  expresses  units  of  the  second  order;  in  the 
third  place,  units  of  the  third  order;  in  the  fourth  place,  units 
of  the  fourth  order;  and  so  on. 

50.  A  digit,  in  any  place  except  the  first,  expresses 
two  values — simple  and  local. 

51.  The  simple  value  of  a  digit  is  the  value  ex- 
pressed by  its  name. 

SS.  The  local  value  of  a  digit  is  the  value  given  to 
it  by  the  place  it  occupies. 


o5 

a 

.2 

73      CO 

-9    S 

0 

.2 

CO 

J 

i 

O 

■4J 

CO 

CO 

1 

M 

^  Names  of  Units, 

rrt      .2        ... 

'Ti 

<v5 

^ 

^ 

rS 

Hundre 
Ten-bill 
Billions 

id 

w 

1 

s 

O 

-a 
p 

1 

Hundre 

Tens. 

Ones, 

ra       rH       rd 

^ 

^ 

rd 

rC 

rP 

ja 

r^      r^       -^ 

(  Places,  and  Or- 

^     +3    "TS 

-1.3 

^ 

-t-3 

-M 

TJ     TS       ixi 

-j              ' 

oq    i-i    o 

T— 1      rH      rH 

Oi 

00 

i^ 

CO 

lO 

'^ 

CO     cq     I— 1 

(      ders  of  Units. 

16    0, 

3 

7 

4, 

5     0     8, 

^ V ' 

Second. 

2     9    4, 

^ V ' 

First.     1 

Figures, 

Fourth. 

Third. 

lumbers.    )    p.^.  ^, 

Billions, 

Millions, 

Thousands, 

O/ies.      Names.       J"  ^  '^'  ^  " 

33.  Notation  is  the  writing  of  numbers. 

34.  Nujneration  is  the  reading  of  numbers. 

35.  Peinciples  of  Notation. 

I.  Ten  units  of  any  order  are  one  unit  of  the  next 
higher  order, 

II.  One  unit  of  any  order  is  ten  units  of  the  next  lower 
order. 


14 


SECOND    BOOK  IN  ARITHMETIC, 


A.  Copy  and  read 


Exercises. 


8,080,800,880 

70,707,070 

999,999,999,999 


4,650,738        232,107,008    800,014,000,200 

93,066,840      3,050,300,125      58,700,025,090 

450,502,000   13,006,010,050    242,424,242,424 

J5.  Write  in  figures 

1.  Eiglit  million  five  thousand  three  hundred  four. 

2.  Eighteen  million  nineteen  ;  fifty-six  thousand. 

3.  Eight  hundred  million  seven  hundred  seven  thou- 
sand five  hundred  six. 

Jf.  Two  hundred  four  million  ninety  thousand  two  hun- 
dred seventeen  ;  fifty  million  thirty  thousand  three. 

5.  Six  hundred  fifty-nine  million  twenty-five. 

6.  Eour  billion  two  hundred  fifty-nine  thousand  thirty. 

7.  Eighty  billion  seven  million  four  hundred  thousand. 

8.  Seventy -four  billion  seven  hundred  four  million 
nine  thousand  ninety-nine. 

9.  Seven  hundred  sixty  billion  eighty  thousand  two 
hundred  ninety-one. 

10.  Three  hundred  billion  seven  hundred  million  two 
hundred  thousand. 

C  Write  in  w^ords 

9,708     72,059,209,000  200,007,000  12,065,587 

400,009                    30,030  80,025,780,240  3,031,050,004 

40,080,276               1,702,050  220,202  135,700,009,015 

26.  The  sign  of  dollars  is  $.  It  is  called  the  dol- 
lar mark. 

A  period  ( . )  is  placed  before  any  number  expressing 
cents.     This  period  is  called  the  decidual  point. 


$38  is  38  dollars. 
$.69  is  69  cents. 
$.07  is  7  cents. 


$5.10  is  5  dollars  ]0  cents. 
$41.08  is  41  dollars  8  cents. 
9,013.44  is  9,013  dollars  44  cents. 


INTEGERS,— NOTATION  AND   NUMERATION.     15 


Copy  and  read  these  num'bers  : 

$759 

$.27 

-   $95.72 

$23,401 

$.19 

$9,450.11 

$107,970 

$.04 

$762,308.05 

$60,060,660 

$.09 

$300.91 

$5,328,153 

$.45 

$8,026,540.00 

S7,  In  expressing  cents,  or  dollars  and  cents,  by  figures, 

1.  Place  the  dollar  marh  hefore  the  number. 

2.  Place  the  decimal  point  before  the  cents. 

3.  Place  a  cipher  between  the  decimal  point  and  any 
numher  of  cents  less  than  10. 

Express  these  numbers  by  figures  : 


Eiglity  dollars  seventy-one  cents. 
Two  hundred  dollars  sixty  cents. 
Nine  thousand  five  dollars  one  cent. 
Four  billion  fifty  dollars  eleven  cents. 
Three  dollars  seventeen  cents. 
Ten  thousand  ten  dollars  nine  cents. 


Twenty  cents. 
Sixty-five  cents. 
Eight  cents. 
Three  cents. 
Ten  cents. 
Fifty-six  cents. 

S8.  EuLE  FOK  Notation. 

Beginning  at  the  left^  write  the  hundreds^  tens^  and 
ones  of  each  period  in  order ;  write  ciphers  in  all  vacant 
places  ;  and  separate  the  periods  by  coinmas. 

29,  Rule  foe  Numeeation. 
Begin  at  the  left  and  read  each  period^  omitting  the 
name  of  the  last  period  and  the  naraes  of  all  places  filled 
with  ciphers. 

ExEECiSEs  IN  Notation  and  Numeeation. 
A.  Copy  and  read 


25,025,025 

600,000,000,000 

15,015,155 

18,018 

3,003 

55,055,000,000 

200,386 

901,020,002,950 

40,425 

8,208 

1,000,890 

25,025 

2,002,002,002 

66,606,066 

260,010,003 

791,056,809 

16  SECOND    BOOK  IN  ARITHMETIC, 

B.  Write  and  read  the  numbers  expressed  by 

1,  6  millions,  3  tens ;  7  ten-thousands,  4  tens,  5  ones. 

^.  1  ten -million,  6  millions,  4  hundred -thousands,  8 
thousands,  5  tens,  2  ones. 

3,  7  hundred-thousands,  3  ten-thousands,  3  hundreds,  7 
tens,  8  ones. 

^.  4  billions,  8  ten-thousands,  3  hundreds,  6  ones. 

5.  3  ten  -  billions,  9  billions,  7  hundred  -  thousands,  9 
thousands. 

6.  4  hundred -billions,  2  ten -billions,  9  ten -millions,  5 
millions,  7  tens. 

C  Express  by  figures 

1.  Thirty-seven  million  ninety-eight  thousand  four  hun- 
dred twenty. 

^.  Eight  million  fifty  thousand;  ninety  million  six 
hundred  thousand  nine  ;  eighty-one  thousand  twelve. 

3.  Six  hundred  one  million  three  hundred  six. 

^.  Nine  billion  two  hundred  seventy  thousand. 

5.  Ninety  dollars  ten  cents ;  twelve  cents. 

6.  Twenty  thousand  six  hundred  seven  dollars  eight 
cents ;  forty-nine  dollars. 

7.  Fifteen  billion  four  million  nine  thousand  eight 
hundred  twenty. 

8.  Six  thousand  one  hundred  one  dollars  five  cents. 

9.  One  hundred  million  twenty -five  thousand  four 
hundred ;  four  hundred  billion  seven  hundred  thousand. 

10.  Nine  hundred  seventeen  billion  three  million  six- 
teen ;  ninety  thousand  five. 

Note  1. — For  outlines  of  notation  and  numeration  for  review,  see  page  269. 

Note  2.— Teachers  who  prefer  that  decimals  should  be  studied  in  con- 
nection with  integers,  will  now  require  their  pupils  to  study  notation  and 
numeration  of  decimals,  pages  111-120. 


SECTION  11. 

ADDITION. 

Oral  Work. — 30.  1.  Eichard  has  3  doves,  and  Thom- 
as has  6.     How  many  doves  have  the  two  boys  ? 

The  t"wo  boys  have  3  doves  and  6  doves,  which  are  9  doves. 

2.  If  you  pay  4  cents  for  figs,  and  7  cents  for  raisins, 
how  many  cents  do  you  pay  out  ? 

3.  James  has  9  marbles,  and  his  brother  George  has  4 
more  than  he  has.     How  many  marbles  has  George  ? 

^.  Sarah  has  9  plums,  and  Emma  has  4.  How  many 
plums  have  the  two  girls? 

6.  A  grocer  sold  5  pounds  of  Japan  tea,  and  8  pounds 
of  black  tea.     How  many  pounds  of  tea  did  he  sell  ? 

6,  Six  boys  and  seven  girls  are  in  the  same  class.  How 
many  pupils  are  in  the  class  ? 

7,  If  you  go  8  miles  east,  and  I  go  9  miles  west  from 
this  school-house,  how  far  apart  shall  we  be  ? 

8.  A  hunter  shot  8  pigeons  and  8  quails.  How  many 
birds  did  he  shoot  ? 

9.  A  man  paid  9  dollars  for  flour,  and  9  dollars  for  coal. 
How  much  did  he  pay  for  both  ? 

31.  A.  To  every  tenth  number,  from  11  to  91  inclusive, 

1.  Add  1.      3.  Add  3.  I  5.  Add  5.      7.  Add  7.        9.  Add    9. 

2.  Add  2.      A,  Add  4.   I   6.  Add  6.      8.  Add  8.       10,  Add  10. 


JB.  To  every  tenth  number,  from  12  to  92  inclusive, 


1.  Add  1. 

2,  Add  2. 


3,  Add  3. 
^.  Add  4. 


5.  Add  5. 

6.  Add  6. 


7.  Add  7. 

8.  Add  8. 


9.  Add    9. 
10.  Add  10. 


18 


SECOND   BOOK   IN  ARITHMETIC. 


€•  To  every  tentli  number,  from  13  to  93  inclusive, 

1.  Add  1.      S.  Add  3.      5.  Add  5.      7.  Add  7.        9.  Add    9. 

2.  Add  2.      4.  Add  4.      ^.  Add  6.      ^.  Add  8.      10,  Add  10. 

3S.  Like  numbers  are  numbers  that  have  the  same 
unit. 

83.  Unlike  nu^nbers  are  numbers  that  have  dif- 
ferent units. 

a.  The  numbers  5  trees,  19  trees,  237  trees  are  like  numbers,  be- 
cause they  have  the  same  unit — 1  tree. 

b.  The  numbers  5  trees,  19  men,  237  dollars  are  unlike  numbers, 
because  they  have  different  units — 1  tree,  1  man,  1  dollar. 

34,  Addition  is  the  process  of  finding  the  sum  of 
two  or  more  like  numbers. 

a.  The  numbers  added  are  the  parts, 

b.  The  number  obtained  by  adding  is  the  sum  or  amount, 

EXEECISES. 

1.  Which  of  the  numbers  5  rods,  16  days,  7  gallons,  3 
days,  9  rods,  12  days,  4  gallons,  13,  6,  87  yards,  are  like 
numbers  ?     Why  ? 

2.  Add  8,  5,  and  3.    Add  6  books,  4  books,  and  Y  books. 

3.  What  is  the  sum  of  9  miles,  and  4  miles,  and  8  miles  ? 
4,.  The  parts  are  4, 11,  and  7.     What  is  the  amount  ? 

5,  The  parts  are  13  bushels,  5  bushels,  and  6  bushels. 
What  is  the  sum  ? 

6,  A  lady  bought  a  bracelet  for  8  dollars,  and  a  locket 
for  15  dollars.     How  much  did  her  purchases  amount  to  ? 

7,  A  cabinet-maker  paid  12  dollars  for  oak  lumber,  and 
9  dollars  for  cherry.     What  sum  did  he  pay  for  lumber  ? 

35.  A,  To  every  tenth  number,  from  14  to  94  inclusive. 


1.  Add  1. 

3,  Add  3. 

5.  Add  5. 

7.  Add  7. 

9.  Add    9. 

^.  Add  2. 

4.  Add  4. 

6,  Add  6. 

S.  Add  8. 

10,  Add  10. 

INTEGERS,— ADDITION. 


19 


B.  To  every  tenth  number,  from  15  to  95  inclusive, 

i.  Add  1.      3.  Add  3.      5.  Add  5.      7.  Add  7.        9.  Add    9. 

2.  Add  2.      4.  Add  4.      ^.  Add  6.      ^.  Add  8.      10.  Add  10. 

30.  The  sign  of  addition  is  an  upright  cross, +• 
It  is  read  and  or  plies ;  plus  means  more. 

87.  The  sign  of  equality  is  two  short,  parallel, 
horizontal  lines  of  equal  length,  ^ . 

4  +  7  +  16  =  27  may  be  read  ''4  plus  7  plus  16  equal  27;"  or  "4 
and  7  and  16  are  27;"  or  "the  sum  of  4,  7,  and  16  is  27." 

A.  Eead— i.  8  +  Y  +  9  +  23==47.   |  ^.  $145 +  $37-^  $182. 
c^.  3  men +  4  men +  17  men  =  24  men. 
^.  21  pounds +-13  pounds  =:  34  pounds. 
6.  8  ounces +  11  ounces +  6  ounces  ==25  ounces. 

Written  WorJ^. — B.  Use  the  proper  signs,  and  write 
L  9, 13,  and  38  are  60. 

2.  7  dollars  plus  16  dollars  equal  23  dollars. 

3.  The  sum  of  7  quarts  plus  15  quarts  plus  9  quarts  is 
31  quarts. 

^.  14  miles,  and  27  miles,  and  7  miles  are  48  miles. 
6.  145  rods  added  to  572  rods  equal  717  rods. 

(7.  Copy  and  complete  each  of  the  following  exercises : 

1.  6  +  8+   2=     \3.     5+   8  +  7  +  4=      \5.  16  +  3  +  2+   4  = 

2.  9  +  4  +  12=      U.   17  +  10  +  3  +  8=      1^.  29  +  5  +  3  +  10  = 

Oral  WorJ^. — 38.  A,  To  every  tenth  number,  from 
16  to  96  inclusive, 


1.  Add  1. 

3.  Add  3. 

5.  Add  5. 

7.  Add  7. 

P.  Add    9. 

2.  Add  2.      4'  Add  4. 

6.  Add  6.      6*.  Add  8. 

10.  Add  10. 

B.  To  every  tenth  number,  from  17  to  97  inclusive. 

1.  Add  1. 

3.  Add  3. 

5.  Add  5. 

7.  Add  7.        P.  Add    9. 

2.  Add  2. 

4.  Add  4. 

^.  Add  6. 

8.  Add  8. 

i(9.  Add  10, 

20 


SECOND    BOOK  IN   ARITHMETIC. 


Written  Work. — 39,  Copy  the  exercises  in  JL,  J5, 
and  C,  and  add  each  exercise  twice,  in  the  order  numbered, 
adding  each  column  first  upward  and  then  downward,  and 
each  Hne  first  from  the  left  and  then  from  the  right. 


A 

1 

2 

3 

4 

5 

10 

II 

12 

13 

14 

6 

2 

8 

5 

3 

1 

15 

3 

7 

1 

6 

4 

7 

6 

3 

9 

6 

2 

16 

8 

2 

6 

1 

2 

8 

9 

7 

4 

1 

4 

17 

5 

7 

3 

9 

8 

9 

2 

1 

8 

5 

7 

18 

3 

4 

9 

5 

4 

B 

1 

2 

3 

4 

5 

12 

13 

14 

15 

16 

17 

6 

8 

4 

6 

5 

7 

18 

5 

3 

9 

0 

4 

2 

7 

3 

9 

0 

2 

8 

19 

8 

2 

8 

3 

5 

3 

8 

6 

5 

3 

7 

4 

20 

7 

1 

7 

8 

5 

4 

9 

7 

2 

5 

8 

9 

2\ 

6 

0 

6 

7 

0 

5 

10 

9 

0 

7 

4 

7 

22 

9 

8 

5 

7 

8 

6 

II 

9 

6 

8 

7 

6 

23 

5 

7 

4 

9 

8 

0 

c 

i_ 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

12 

13 

7 

6 

9 

4 

3 

8 

2 

5 

6 

2 

4 

8 

14 

4 

8 

3 

9 

9 

6 

5 

7 

2 

8 

8 

3 

15 

9 

5 

8 

9 

5 

8 

4 

6 

5 

9 

9 

5 

16 

8 

4 

7 

9 

8 

8 

9 

3 

7 

9 

8 

0 

n 

1 

0 

9 

9 

6 

8 

3 

2 

3 

9 

7 

4 

18 

6 

5 

7 

8 

2 

8 

2 

9 

7 

9 

6 

9 

!i 

5 

3 

6 

8 

4 

8 

0 

7 

6 

5 

6 

3 

20 

4 

9 

7 

8 

4 

8 

7 

6 

7 

9 

4 

6 

2\ 

1 

5 

6 

7 

3 

8 

6 

8 

4 

9 

3 

0 

22 

8 

2 

7 

7 

9 

8 

9 

2 

7 

9 

2 

5 

23 

9 

0 

4 

7 

9 

8 

8 

8 

2 

5 

9 

8 

24 

7 

9 

7 

6 

7 

8 

7 

4 

7 

9 

0 

2 

25 

7 

7 

5 

6 

8 

8 

2 

9 

9 

9 

8 

3 

26 

8 

8 

7 

6 

8 

8 

2 

3 

7 

9 

9 

9 

27 

6 

5 

2 

0 

7 

3 

8 

9 

7 

8 

7 

3 

INTEGERS.— ADDITION. 


21 


40.  Copy,  complete,  learn,  and  recite  tjie 

Table  of  Primary  Combinations  in  Addition. 


1+1= 

2  +  1  = 

3  +  1  = 

4  +  1  = 

8  +  1  = 

7  +  1  = 

2  +  2  = 

3  +  2  = 

7  +  2  = 

6  +  2  = 

6  +  1  = 

5  +  1  = 

6  +  3  = 

5  +  3  = 

5+2  = 

4  +  2  = 

5  +  4  = 

4  +  4  = 

4  +  3  = 

3  +  3  = 

9+1  = 

9  +  2  = 

9  +  3  = 

9  +  4  = 

8  +  2  = 

8  +  3  = 

8  +  4  = 

8+5  = 

7  +  3  = 

7  +  4  = 

7  +  5  = 

7  +  6  = 

6  +  4  = 

5  +  5  = 

6+5  = 

9  +  7  = 

6  +  6  = 
9  +  6  = 

9  +  5  = 
8  +  6  = 

9  +  8  = 

8  +  8  = 

8  +  7  = 

7  +  7  = 

9  +  9  = 

Note.  —This  table  contains  all  the  combinations  that  can  be  formed  bj 
adding  any  two  of  the  first  nine  integers. 

41.  Exercises  in  Addition  at  Sight. 
519483726 
111111111 


6 


Note.— The  preceding  exercises  are  to  be  written  upon  the  board,  and 
used  in  class  drill  daily,  until  every  pupil  can  give,  at  sight,  the  sum  of  the 
numbers  expressed  by  any  two  digits. 


22 


SECOND    BOOK  IN  ARITHMETIC, 


Oral  Work. — 4S.  A.  To  every  tenth  number,  from 
18  to  98  inclusive, 


1,  Add  1. 

^.  Add  3. 

5.  Add  5. 

7.  Add  7. 

P.  Add    9. 

2.  Add  2. 

^.  Add  4.      6,  Add  6. 

<^.  Add  8. 

10.  Add  10. 

B.  To  every  tentli  number,  from  19  to  99  inclusive, 

i.  Add  1. 

^.  Add  3. 

5.  Add  5.       7.  Add  7. 

P.  Add    9. 

2,  Add  2. 

Jf.  Add  4. 

<?.  Add  G.       ^.  Add  8. 

10.  Add  10. 

€•  To  every  tenth  number,  from  10  to  100  inclusive. 

1.  Add  1. 

^.  Add  3. 

5.  Add  5. 

7.  Add  7. 

9.  Add    9. 

2,  Add  2. 

Jf.  Add  4. 

^.  Add  6. 

8.  Add  8. 

i^  Add  10. 

43.  A.  (a)  15  days,  230  days,  9  rods,  70  pounds,  16 
bushels,  50  days,  37  rods. 

(h)  194  pears,  $3,115,  280,  $16,  75  pears,  $36,  283  pears. 
(c)  68  boys,  13  horses,  291  sheep,  8  houses,  1,751  pigeons. 

1.  Which  numbers  in  (a)  can  be  added  ?    Why  ? 

2.  Wliich  numbers  in  (b)  can  be  added  ?     Why  ? 

3.  Which  numbers  in  (h)  can  not  be  added  ?    Why  ? 
^.  Which  numbers  in  (c)  can  be  added  ?    Wliy  ? 

5.  What  numbers  can  be  added  ? 

6.  What  numbers  can  not  be  added  ? 
J3,  1.  Ones  must  be  added  to  what  ? 

2.  Tens  must  be  added  to  what  ?  Hundreds,  to  what  ? 
^.  To  what  can  a  unit  or  units  of  any  order  be  added  ? 
4-  To  what  can  not  a  unit  or  units  of  any  order  be  added  ? 

5.  Add  300,  200,  700,  500,  and  100. 

6.  Add  $.10,  $.90,  $.30,  $.50,  $.80,  and  $.60. 

7.  Add  7,000,  1,000,  4,000,  2,000,  and  5,000. 
C  How  many  ones  and  how  many  tens  are  there 

1.  In  the  sum  of  7,  5,  8,  and  3  ? 

2.  In  the  sum  of  4,  9,  6,  2,  and  1  ? 
S.  In  the  sum  of  10,  30,  20,  and  40  ? 


y  Parts. 


INTEGERS.— ADDITION.  23 

44.  Pkijs-ciples  of  Addition. 
I.  A  whole  equals  the  sum  of  all  its  parts, 
II.  Only  units  of  like  orders  can  he  added. 

Written  Work.  — ^5.  Ex.  The 

parts  are  720,  4,813,  1,327,  504,  and 
6,732.     "What  is  their  sum  ? 

Explanation. — I  write  the  parts — ones 
under  ones,  tens  under  tens,  and  so  on — 
and  draw  a  horizontal  line  below  the 
last  number. 

Addinf?  the   ones,  I   have  16  (=  6  ones         'i~i~r^~n~n  ^ 
A  -i   4-     \        JT        -4.1      n  1  lA.UiJO    Sun 

and  1  ten),  and  1  write  the  6  ones  be-  ^' 

low  the  line,  for  the  ones  of  the  required 

sum.  Adding  the  1  ten  with  the  tens  of  the  given  numbers,  I 
have  9  tens,  which  I  write  for  the  tens  of  the  sum.  Adding  the 
hundreds,  I  have  30  (=  3  thousands),  and  I  WTite  0  in  the  place 
of  hundreds  in  the  sum.  Adding  the  3  thousands  with  the  given 
thousands,  I  have  14  thousands  {-  4  thousands  and  1  ten-thou- 
sand), which  I  write  for  the  thousands  and  ten-thousand  of  the 
sum. 
The  result,  14,096,  is  the  required  sum. 


Process. 

720] 

i,813 

1,827 

SOU 

6,782  J 

PROBLEMS. 

A  1 

2 

3 

4- 

5 

657 

$3,140 

36 

$52.41 

4,213 

360 

1,205 

725 

.78 

145 

208 

4,332              1,413 

48.36 

32 

845 

2,011              9,879 

9.29 

5,276 

$ 

6 

7 

8 

9 

$100,872 

125,600 

$ 

132.79 

5,267 

63,494 

85,638 

31.48 

221,532 

926,735 

7,257 

238.40 

99,835 

19,622 

735,528 

9,478.43 

892,763 

5,368 

863,264 

68.57 

81,676 

824,463 

856,470 

2.56 

725,052 

24 


SECOND    BOOK  IN  ARITHMETIC. 


J3.  Copy  these  exercises  and  add  each  one  twice,  in  the 
order  numbered,  adding  each  column  first  upward  and 
then  downward,  and  each  line  first  from  the  left  and  then 


from  the  right. 


1 

2 

3 

4 

10 

II 

12 

13 

$20 

$15 

$30 

$12 

•4  100 

333 

350 

116 

$19 

$10 

$17 

$44 

15  321 

240 

175 

210 

$55 

$27 

$13 

$36 

16  105 

•  500 

920 

800 

$32 

$16 

$11 

$23 

17  621 

127 

155 

513 

$40 

$21 

$34 

$18 

18  584 

321 

230 

109 

19 

20 

21 

22 

23 

24 

25  $25.30 

$  9.37 

$60 

$88.94 

$75.30 

$  .08 

26  $18.05 

$28.25 

$  3.55 

$73 

$  .04 

$15.15 

27  $50.15 

$  .09 

$  .90 

$16.91 

$59.50 

$11.11 

28  $  .75 

$10.06 

$35.16 

$  .07 

$65.95 

$  8 

29  $36 

$49.29 

$70 

$66.10 

$  4 

$17.40 

30  $40.10 

$  5.03 

$12.22 

$  4 

$97.10 

$  9.09 

Oral  Work. — C  1.  Albert  has  19  apples,  Frank  has 
5,  Edgar  has  7,  and  Charles  has  10.  How  many  apples 
have  the  four  boys  ? 

The  four  boys  have  the  sum  of  19  apples,  5  apples,  7  apples, 
and  10  apples,  which  is  41  apples. 

2,  One  winter  a  farmer  fed  14  tons  of  hay  to  his  cows, 
5  tons  to  his  horses,  and  9  tons  to  his  sheep.  How  much 
hay  did  it  take  to  winter  his  live  stock  ? 

3.  A  farmer  sowed  20  acres  of  land  to  oats,  30  acres  to 
wheat,  and  10  acres  to  corn.    How  many  acres  did  he  sow  ? 

^.  A  boy  gave  60  cents  for  an  arithmetic  and  20  cents 
for  a  slate.     How  much  did  he  give  for  both  ? 

5.  What  is  the  sum  of  66,  5,  6,  and  9  ? 

6.  How  many  are  48  +  3  +  6  +  9  +  4? 

7.  What  is  the  amount  of  $32,  $9,  $6,  $4,  and  $7? 


INTEGERS.— ADDITION.  25 

8,  I  paid  30  cents  for  a  pound  of  coffee,  and  14  cents 
for  a  quart  of  molasses.     How  mucli  did  I  pay  for  both  ? 

I  paid  the  sum  of  30  cents  and  14  cents ;  14  cents  are  10 
cents -f-  4  cents  ;  30  cents  and  10  cents  are  40  cents,  and  4  cents 
are  44  cents.    Hence,  I  paid  44  cents  for  both. 

9.  A  grazier  has  70  head  of  cattle  in  one  pasture,  and 
41  head  in  another.     How  many  cattle  has  he  ? 

10,  A  shoe  dealer  sold  40  pairs  of  kip  boots,  and  38 
pairs  of  calf  boots.    How  many  pairs  of  boots  did  he  sell  ? 

11.  A  hatter  bought  70  cases  of  silk  hats,  and  66  cases 
of  felt  hats.     How  many  cases  of  hats  did  he  buy  ? 

12,  A  clothier  sold  30  sack  coats,  40  dress  coats,  and  18 
overcoats.     How  many  coats  did  he  sell  ? 

13.  How  many  books  are  40  books  +  20  books  4-  56  books  ? 
U.  What  is  the  amount  of  $  .80,  %  .30,  $  .40,  and  $  .25  ? 
15,  What  is  the  sum  of  200,  700,  300,  and  250  ? 

Written  Work. 
Z>  L  -  1  -  - 

7,887  600,954    $  4,718.8.5    $5,109.32  $  9,800.25 

7,538  432,798    28,569.09       84.69  16,000.10 

6,852  759,890     1,296.52      730  44,999.87 

3,929  635,428     54,903.28        8.05  65,874.08 

4,387  970,325     4,925       6,043.17  3.75 

25            $          $  7,444.15 

8,110          5             6  68,974 

478       1,325,086      1,368,122,097  13.64 

9        764,572      4,513,265,928  98,525.23 

808       3,426,515      9,157,934,684  .09 

2,491       1,812,328      1,369,668,624  400.45 

6      25,680,087      7,117,944,958  22,776 

374      7,548,766       461,371,480  18,106.30 

9,466       6,754,645       804,875,333  78,003.14 

$ 

B 


26  SECOND    BOOK  JN  ARITHMETIC. 

Oval  Work, — E.  1,  Alva  paid  21  cents  for  a  knife, 

and  18  cents  for  a  ball.     How  much  did  both  cost  him  ? 

Both  cost  him  the  sum  of  21  cents  and  18  cents;  21  cents 
and  10  cents  are  31  cents,  and  8  cents  are  39  cents.  Hence, 
both  cost  him  39  cents. 

2.  Helen  paid  63  cents  for  butter,  and  31  cents  for 
cheese.     How  much  did  her  purchases  amount  to  ? 

3,  A  lady  paid  48  dollars  for  a  book-case,  and  12  dol- 
lars for  a  table.     How  much  did  she  pay  for  both  ? 

Jf..  Enos  gave  $  .45  for  a  sled,  and  $  .18  to  have  it 
painted.     How  much  did  the  sled  cost  him  ? 

6.  A  merchant  sold  one  cloak  for  $28,  and  another  for 
$27.     How  much  did  he  receive  for  both  ? 

6.  Lucy  paid  $  .38  for  a  pair  of  gloves,  and  $  .25  for  a 
comb.     How  much  did  she  pay  for  all  ? 

7.  A  coal  dealer  bought  90  tons,  73  tons,  and  85  tons 
of  coal.     How  many  tons  did  he  buy  ? 

8.  Upon  a  steamboat  are  85  men,  64  women,  and  13 
children.     How  many  persons  are  on  the  boat  ? 

9.  How  many  horses  ar^  41  horses,  98  horses,  and  11 
horses  ? 

10.  $34  +  $27-f-$45=how  many  dollars? 

11.  56  sheep +  18  sheep +  47  sheep = how  many  sheep? 
m.  What  is  the  sum  of  44,  37, 16,  and  7  ? 

13.  The  parts  are  59,  34,  83,  and  12.    Find  the  amount. 

Written  Work.  —  Fi  1.  What  is  the  sum  of  123 

pounds,  103  pounds,  and  3,140  pounds  ? 

2.  What  is  the  amount  of  $96.12,  $132.07,  $98.76, 
$72.38,  and  $115.25  ? 

3.  Add  $7.28,  $241.09,  $  .42,  $  .96,  and  $44.52. 

Jf.  A  miller  bought  1,284  bushels  of  wheat,  and  859 
bushels  of  corn.    How  many  bushels  of  grain  did  he  buy  ? 


INTEGERS.— ADDITION.  27 

5.  The  four  quarters  of  an  ox  weighed  142  pounds,  137 
pounds,  181  pounds,  and  184  pounds.  What  was  their 
total  weight  ? 

6.  In  five  days  a  dairyman  made  37  pounds,  42  pounds, 
34  pounds,  55  pounds,  and  43  pounds  of  butter.  How 
many  pounds  of  butter  did  he  make  in  the  five  days  ? 

7.  I  sold  a  wash-stand  for  $6.50,  a  bureau  for  $11.63,  and 
an  easy-chair  for  $8.25.    How  much  did  I  receive  for  them  ? 

8.  One  week  a  railroad  company  bought  154  cords,  115 
cords,  180  cords,  145  cords,  94  cords,  and  269  cords  of 
wood.     How  many  cords  were  bought  that  week  ? 

9.  A  merchant  gained  $218  on  a  lot  of  goods  that  cost 
him  $463.     For  how  much  did  he  sell  the  goods  ? 

10.  Three  men.  A,  B,  and  C,  buy  a  machine-shop,  A  fur- 
nishing $2,163  of  the  purchase  money,  B  $3,085,  and  C 
$1,236.     How  much  does  the  shop  cost  ? 

11.  A  lumberman  drove  seven  rafts  of  logs  down  Pe- 
nobscot Eiver  to  Bangor.  In  the  *first  raft  were  1,276 
logs,  in  the  second  859,  in  the  third  1,009,  in  the  fourth 
793,  in  the  fifth  1,318,  in  the  sixth  925,  and  in  the  seventh 
1,158.     How  many  logs  were  in  all  the  rafts  ? 

12.  I  bought  a  village  lot  for  $325,  and  paid  $22.63  for 
taxes.     For  how  much  must  I  sell  it,  to  gain  $72.37? 

46,  EuLE  FOR  Addition  of  Integees. 

I.  Write  the  numhers  so  that  units  of  the  same  order 
may  he  in  the  same  column. 

II.  Add  the  units  of  the  lowest  order ^  write  the  ones  of 
the  sum,  in  the  result^  and  add  the  t&ns  of  the  sum  with 
the  units  of  the  next  order. 

III.  So  proceed  with  the  units  of  each  order ^  and  write 
the  entire  sum  of  the  units  of  the  highest  order. 


28  SECOND    BOOK  IN  ARITHMETIC, 

Pkoblems. 
A.  1.  Add  346,  5,279,  and  8,165. 

^.  What  is  the  sum  of  376,  128,  593,  and  842  ? 

3.  $2,317,  $954,  $1,683,  $75,  and  $381  amount  to 
how  many  dollars? 

4..  Find  the  sum  of  56,  1,642,  485,  691,  3,482,  and 
22,849. 

5.  What  is  the  amount  of  $.93,  $.25,  $.81,  $.19, 
$  .44,  $  .75,  and  $  .50  ? 

6.  Add  720,  4,813,  1,327,  504,  and  6,732. 
Wkitten  Pkocess.       Making  the  Computation. 

720  6,  13,  16  ;  write  6  (add  1). 

4^813  4,  6,  7,  9;  write  9. 

1,32  7  12,  15,  23,  30  ;  write  0  (add  3). 

gQJ^^  9,  10,  14;  write  14. 
6  732  Result,  14,096. 

lJf.,096 

7.  A  lady  paid  $25  for  a  bureau,  $42  for  6  chairs,  $65 
for  a  sofa,  $57  for  three  tables,  and  $35  for  a  bedstead. 
How  much  did  her  purchases  amount  to  ? 

8.  One  forenoon  a  street  car  made  5  trips,  carrying  54 
passengers  the  fii-st  trip,  48  the  second,  63  the  third,  62 
the  fourth,  and  49  the  fifth.  How  many  passengers  did 
the  car  carry  that  forenoon  ? 

9.  A  fruit  grower  sold  74  barrels  of  russet  apples,  56 
barrels  of  greenings,  and  152  barrels  of  pippins.  How 
many  barrels  of  apples  did  he  sell  ? 

10.  How  many  rods  of  fence  will  it  take  to  inclose  a 
field  that  is  43  rods  long  and  24  rods  wide  ? 

11.  In  nine  days  I  travel  60  miles,  110  miles,  35  miles, 
69  miles,  86  miles,  94  miles,  115  miles,  25  miles,  and  100 
miles.     How  many  miles  do  I  travel  ? 


INTEGERS.— ADDITION.  29 

12.  A  store  on  the  first  floor  of  a  building  rents  for 
$1,365  a  year,  the  offices  on  the  second  floor  rent  for  $762, 
and  a  photograph  room  on  the  third  floor  rents  for  $278. 
How  much  is  the  whole  rent  of  the  building  ? 

13.  A  produce  dealer  bought  720  bushels,  145  bushels, 
and  1,124  bushels  of  oats.  How  many  bushels  of  oats  did 
he  buy  ? 

Oral  Work* — S,  1.  In  an  orchard  are  20  pear-trees, 
10  more  apple-trees  than  pear-trees,  15  plum-trees,  and  6 
more  cherry-trees  than  plum-trees.  How  many  trees  are 
in  the  orchard? 

2.  A  Swede  came  to  America  when  he  was  11  years 
old  ;  14  years  afterward  he  married  ;  6  years  later  his  wife 
died,  and  she  has  been  dead  13  years.     What  is  his  age  ? 

S.  In  a  park  are  an  elm  40  feet  high,  an  oak  50  feet 
higher  than  the  elm,  and  a  pine  60  feet  higher  than  the 
oak.     What  is  the  height  of  each  tree  ? 

^.  A  nurseryman  sold  60  pear-trees  to  one  farmer,  30 
to  another,  and  17  to  another.  How  many  pear-trees  did 
he  sell? 

5.  A  farmer  has  14  cattle,  11  horses,  and  23  sheep. 
How  many  head  of  live  stock  has  he  ? 

6.  In  building  a  bridge  it  took  64  days  to  do  the  mason 
work,  46  days  the  wood  work,  and  18  days  the  grading. 
How  many  days  did  it  take  to  build  the  bridge  ? 

7.  I  paid  84  cents  for  fruit,  and  66  cents  for  candies. 
How  much  did  I  pay  for  all  ? 

8.  James  paid  25  cents  for  a  slate,  and  37  cents  for  a 
reader.     How  many  cents  did  he  pay  for  both  ? 

9.  A  lady  taught  a  summer  term  of  65  days,  and  a 
winter  term  of  76  days.  How  many  days  did  she  teach 
in  the  year  ? 


30  SECOND    BOOK  IN  ARITHMETIC. 

10.  My  COWS  give  49  quarts  of  milk  in  the  morning, 
and  67  quarts  in  the  evening.  How  many  quarts  do  they 
give  in  a  day  ? 

11,  Orson  found  28  plums  under  one  tree,  and  94  plums 
under  another.     How  many  plums  did  he  find  ? 

1%.  A  steamship  landed  in  New  York  with  45  first-class, 
16  second-class,  and  72  steerage  passengers.  How  many 
passengers  did  she  bring  ? 

13,  Louise  is  15  years  old,  her  mother  is  27  years  older 
than  she,  and  her  grandmother  is  35  years  older  than  her 
mother.     How  old  is  her  grandmother  ? 

7^.  A  farmer  has  28  acres  of  woodland,  40  acres  of 
pasture,  30  acres  of  meadow,  16  acres  in  wheat,  3  acres  in 
potatoes,  4  acres  in  corn,  20  acres  in  oats,  2  acres  in  root 
crops,  and  1  acre  in  yard  and  garden.  How  many  acres 
are  there  in  his  farm? 

Written  Work.-^C.  1,  A  real-estate  agent  sold  two 
houses  for  $1,830  each,  a  city  lot  for  $2,210,  and  a  farm 
for  $12,125.     How  much  did  the  sales  amount  to  ? 

^.  C  has  $4,258,  B  has  $328  more  than  C,  A  has  $529 
more  than  B,  and  D  has  as  much  as  B,  C,  and  A.  How 
much  money  have  the  four  men  ? 

3.  What  is  the  sum  of  thirty-five  million  eight  hundred 
seventy-six  thousand  one  hundred  twenty,  three  hundred 
ninety-six  thousand  four  hundred  ninety-one,  and  five  hun- 
dred forty-three  thousand  six  hundred  seven  ? 

^.  A  man  bought  three  houses,  paying  for  them  $1,804, 
$1,602,  and  $2,355.  He  sold  them  for  $2,395  above  cost. 
How  much  did  he  receive  for  them? 

5.  A  fruit  dealer  paid  $15.45  for  oranges,  $20.34  for 
lemons,  $27.59  for  pine-apples,  and  $16.72  for  cocoa-nuts. 
How  much  did  the  fruit  cost  him  ? 


INTEGERS.— ADDITION.  31 

6.  A  man  paid  $3,256  for  a  farm,  $1,217  for  the  live 
stock,  $557  for  farming  implements,  $373  for  the  crops 
on  the  ground,  and  $439  for  repairs.  How  much  was^his 
total  outlay  ? 

7.  A  contractor  received  $14,100  for  building  a  hotel, 
$885  for  building  a  livery  stable,  $2,637  for  building  a 
house,  and  $3,233  for  building  a  warehouse.  How  much 
did  he  receive  for  the  four  jobs  ? 

8.  Monday  I  deposited  $52.18  in  the  bank,  Tuesday 
$68.47,  Wednesday  $45.52,  Thursday  $78.77,  Friday  $80.16, 
and  Saturday  $119.82.  What  was  the  amount  of  my  de- 
posits for  the  week  ? 

9.  One  month  a  grocer  received  $486.69  for  sugars, 
$390.25  for  teas,  $166.50  for  fruits,  and  $1,767  for  other 
goods.     What  was  the  amount  of  his  sales  for  the  month  ? 

10.  Find  the  sum  of  15  million  9  thousand  17,  9  mill- 
ion 508,  675  thousand  899,  and  245  million  326  thou- 
sand 8. 

11.  North  America  contains  8,593,000  square  miles. 
South  America  7,362,000  square  miles,  Europe  3,825,000 
square  miles,  Asia  17,300,000  square  miles,  and  Africa 
11,557,000  square  miles.  How  many  square  miles  are 
there  in  these  five  continents? 

12.  In  seven  piles  of  wood,  measuring  364  cords,  729 
cords,  95  cords,  832  cords,  723  cords,  407  cords,  and  634 
cords,  are  how  many  cords  ? 

13.  A  butcher  bought  five  oxen,  which  weighed  1,120 
pounds,  1,312  pounds,  1,250  pounds,  1,547  pounds,  and 
1,420  pounds.     What  was  the  total  weight  ? 

U.  What  is  the  sum  of  $1,328,654,  $6,863,  $11,496, 
$2,000,342,  $658,  $9,891,  $78,620,  $1,060,909,  $5,683, 
$22,408,  $23,684,901,  $846,  $29,  and  $4,803? 


32  SECOND    BOOK  IN  ARITHMETIC. 

15.  A  mercliant  tailor  paid  $315.Y8  for  broadcloths, 
$282.25  for  cassimeres,  $106.45  for  vestings,  $24.90  for 
linings,  and  $14.04  for  thread,  silk,  and  buttons.  How 
much  did  he  paj  out  for  stock  ? 

16.  What  is  the  amount  of  $58.48,  $368,  $37.28, 
$9.99,  $48.06,  $768.04,  $362.22,  and  $8.37? 

17.  A  druggist  pays  $1,335  a  year  for  rent,  $2,447  for 
clerk  hire,  $292.25  for  fuel,  $259.19  for  gas,  $875  for 
freight  and  cartage,  and  $1,936  for  other  expenses.  How- 
much  are  his  yearly  expenses  ? 

18.  A  merchant's  cash  sales  Monday  were  $98.19,  Tues- 
day $132,  Wednesday  $198.07,  Thursday  $272.63,  Friday 
$115.01,  and  Saturday  $295.  What  was  the  amount  of 
his  cash  sales  for  the  week  ? 

19.  A  farmer  raised  1,286  bushels  of  wheat,  229  bush- 
els of  corn,  144  bushels  of  buckwheat,  683  bushels  of  oats, 
257  bushels  of  rye,  and  599  bushels  of  barley.  How  many 
bushels  of  grain  did  he  raise  ? 

20.  A  owes  B  $901.32,  C  $321.09,  D  $288.58,  E  $124.15, 
F  $150.75,  and  G  $50.90.     How  much  does  he  owe  ? 

21.  What  is  the  sum  of  ninety-nine  thousand  ninety- 
nine,  twenty-seven  million  five  hundred,  forty-two  million 
two  thousand  five,  four  hundred  eight  thousand  ninety-six, 
and  five  thousand  four  hundred  thirty-seven  ? 

22.  Find  the  amount  of  nine  hundred  seven  million 
eight  hundred  five  thousand  seventy-four,  one  thousand 
nine  hundred  fifty,  twenty-four  million  twenty-four,  seven 
million  eight  hundred  nineteen  thousand,  six  hundred 
twelve,  and  one  hundred  fifty-seven. 

Note  1. — For  outlines  of  addition  for  review,  see  page  269. 

Note  2. — Teachers  who  prefer  that,  decimals  should  be  studied  in  con- 
nection with  integers,  will  now  require  their  pupils  to  study  addition  of 
decimals,  pages  121, 122. 


SECTION   III. 

SUBTRACTION. 

Oral  Work.— ^7.  1,  On  a  tree  were  8  peaches,  but 
Laura  picked  5  of  them.  How  many  peaches  were  left 
on  the  tree  ? 

There  vrere  left  on  the  tree  8  peaches  less  5  peaches,  -which 
are  3  peaches. 

2.  James  has  7  rabbits ;  4  of  them  are  gray,  and  the 
others  are  white.     How  many  white  rabbits  has  he  ? 

3.  A  man  bought  9  pounds  of  butter,  and  in  a  week  his 
family  used  all  but  3  pounds  of  it.  How  many  pounds  of 
butter  were  used  ? 

J/..  If  a  factory  girl  earns  $12  in  a  week,  and  pays  $3  for 
board,  how  much  money  has  she  left  ? 

5.*  I  paid  $13  for  a  silver  cake  basket,  and  $4  less  for  a 
set  of  tea-spoons.     How  much  did  the  spoons  cost  me  ? 

6.  A  boy  bought  a  water-melon  for  15  cents,  and  a 
bunch  of  grapes  for  6  cents.  How  much  more  did  he  pay 
for  the  melon  than  for  the  grapes  ? 

7.  A  farmer  having  16  cow^s,  sold  7  of  them.  How 
many  did  he  keep  ? 

8,  Nelson  is  17  years  old ;  how  old  was  he  9  years  ago  ? 

9,  There  are  6  passengers  in  a  street  car  which  has 
seats  for  18  passengers.     How  many  seats  are  vacant  ? 

48.  A.  From  every  tenth  number,  from  11  to  91  in- 
clusive. 


1.  Subtract  1. 
^.  Subtract  2. 


3.  Subtract  8. 
Jf.  Subtract  4. 
5.  Subtract  5. 


6.  Subtract  6. 

7.  Subtract  7. 

8.  Subtract  8. 
B  2 


9.  Subtract    9. 
10.  Subtract  10. 


34 


SECOND    BOOK  IN  ARITHMETIC. 


3.  From  every  tenth  number,  from  12  to  92  inclusive, 


1.  Subtract  1. 

3.  Subtract  3. 

6.  Subtract  6. 

9.  Subtract    9, 

2.  Subtract  2. 

Jf.  Subtract  4. 

7.  Subtract  1. 

10.  Subtract  10. 

5.  Subtract  5. 

8.  Subtract  8. 

C  From  every  tenth  number,  from  13  to  93  inclusive. 

1,  Subtract  1. 

S,  Subtract  3. 

6.  Subtract  6. 

9.  Subtract    9, 

2.  Subtract  2. 

Jf.  Subtract  4. 

7.  Subtract  1. 

10.  Subtract  10. 

5.  Subtract  5. 

8.  Subtract  8. 

49.  Stibtraction  is  the  process  of  taking  one  of  two 
like  numbers  from  the  other. 

a.  That  one  of  the  two  numbers  from  which  the  other  is 

to  be  taken  is  the  minuend, 
6.  The  number  to  be  taken  from  the  minuend  is  the  «m5- 

trahend, 
c.  The  number  obtained  by  subtracting  is  the  difference 

or  remainder. 

Exercises. 
1.  Subtract  7  books  from  11  books. 
^.  Subtract  9  pins  from  15  pins. 

3.  What  is  the  difference  between  13  horses  and  4  horses  ? 
J/..  Take  8  leaves  from  17  leaves.   What  is  the  remainder  1 

5.  If  you  subtract  6  cents  from  15  cents,  what  is  the 
remainder  ? 

6.  The  minuend  is  14,  and  the  subtrahend  is  5.     What 
is  the  difference  ? 

7.  A  cook  having  16  eggs,  used  9.     How  many  had  she 
left?  " 

8.  In  question  7,  which  number  is  the  minuend  ?  Which 
is  the  subtrahend  ?     Which  is  the  remainder  ? 

9.  In  a  garden  is  one  vase  worth  $19,  and  another  worth 
$14.     What  is  the  difference  in  their  values  ? 


INTEGERS.— 8UBTRA  CTION. 


35 


10.  In  going  12  miles,  I  walked  5  miles,  and  rode  the 
remainder  of  the  distance.     How  many  miles  did  I  ride  ? 

11.  The  minuend  is  $  .25,  and  the  subtrahend  is  %  .15. 
What  is  the  difference  ? 

50,  A.  From  every  tenth  number,  from  14  to  94  in- 
clusive, 


1.  Subtract  1. 

2.  Subtract  2. 


3.  Subtract  3. 
Jf..  Subtract  4. 
5.  Subtract  5. 


Subtract  6. 
Subtract  7. 
Subtract  8. 


9.  Subtract    9. 
10.  Subtract  10. 


J5.  From  every  tenth  number,  from  15  to  95  inclusive, 

1.  Subtract  1.     3.  Subtract  3.     6.  Subtract  6.       9.  Subtract    9. 

2.  Subtract  2.     ^.  Subtract  4.     7.  Subtract  Y.     10.  Subtract  10. 

5.  Subtract  5.     8.  Subtract  8. 

51.  The  sign  of  subtraction  is  a  short,  horizontal 
line,  — .    It  is  read  mimis  or  less ;  minus  means  less. 

25  — 16  may  be  read  "25  minus  16,"  or  "25  less  16." 
A.  Read  these  exercises : 


Jf.  $31.10-$9.75  =  $21.35 

5.  32  men  — 25  men  =  7  men. 

6.  50  pounds— 32  pounds=:18  pounds. 
Written  Work. — S.  Write  each  of  these  exercises, 

using  the  proper  signs  : 


23-9  =  14 

$182  — $37i=$145 
$.87  — $.31  =  $. 56 


1.  14:  from  21  leaves  7. 

^.15  guns  less  9  guns  equal  6 

guns. 
3.  2^  men  minus  16  men  equal 

7  men. 


4.  $525  less  $500  are  $25. 

5.  48  miles  from  60  miles  equal 

12  miles. 
^.15  rods  taken  from  145  rods 
leave  130  rods. 


C.  Copy  and  complete  these  exercises  : 


1.  19-   4  = 

3.    17-9  = 

2.  41—21  = 

4.  $19-$4  = 

S.  41  lemons— 21  lemons= 

^.17  books  — 9  books  = 

7.  36  yards- ^ 

5  yards  = 

25  fishes  — 17  fishes = 
9.  40  cars  — 31  cars= 

10.  32  bricks— 8  bricks  = 

11.  51  steps— 40  steps  = 

12.  100  cents— 90  cents  = 


bricks, 
steps, 
cents.. 


13.  43  dollars  — 30  dollars = 


dollars. 


86 


SECOND    BOOK  IN  ARITHMETIC. 


Oral  Work.  —  5S.  A.  From  every  tenth  number, 
from  16  to  96  inclusive, 

1.  Subtract  1.  I  3.  Subtract  3.     6.  Subtract  6.       9.  Subtract    9. 

2.  Subtract  2.     Jf.  Subtract  4.     7.  Subtract  7.     10.  Subtract  10. 

5.  Subtract  5.     8.  Subtract  8. 

3.  From  every  tenth  number,  from  lY  to  97  inclusive. 


1.  Subtract  1. 

2.  Subtract  2 


3.  Subtract  3. 
J/,.  Subtract  4. 
5.  Subtract  5. 


6.  Subtract  6. 

7.  Subtract  7. 

8.  Subtract  8. 


9.  Subtract    9. 
10.  Subtract  10. 


Written  Work.  —  53.  Copy,  complete,  learn,  and 
recite  the 

Table  of  Primary  Combinations  in  Subtraction. 


2  —  1  = 

3-1  = 

4-1  = 

5-1  = 

6-1  = 
6-5  = 
6-2  = 

3  —  2  = 

7-1  = 
7-6  = 

4-3  = 
4  —  2  = 

8-1  = 

5  —  4  = 
5  —  2  = 
5-3  = 

6—4  = 

7-2  = 

8-7  = 

9  —  1  = 

6-3  = 

7-5  = 

8-2  = 

9-8  = 

10  —  1  = 
10-9  = 
10-2  = 
10-8  = 
10-3  = 
10-7  = 

7-3  = 

7-4  = 

11-2  = 
11—9  = 
11—3  = 
11-8  = 

8-6  = 
8-3  = 
8-5  = 
8-4  = 

12-3  = 
12  —  9  = 

9  —  2  = 
9-7  = 
9  —  3  = 
9-6  = 
9—4  = 
9-5  = 

10  —  4  = 

11—4  = 

12—4  = 

13-4  = 

10  —  6  = 

11-7  = 

12  —  8  = 

13  —  9  = 

10-5  = 

11-5  = 

12-5  = 

13  —  5  = 

14  —  5  = 
14-9  = 
14  —  6  = 

11  —  6  = 

15-6  = 
15  —  9  = 

12-7r^ 

12-6  = 
16-7  = 

13-8  = 
13-6  = 
13-7  = 

14  —  8  = 

15-7c= 

16-9  = 

17-8  = 

14-7  = 

15-^^ 

16-8  = 

17-9= 

18-9  = 


INTEGERS.-^SUBTRACTION. 


37 


54.  Exercises  in  Subtraction  at  Sight. 


M 


1 

4 

7 

10 

2 

5 

8 

3 

6 

9 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

2 

1 

3 

8 

4 

9 

5 

10 

6 

11 

2 

2 

2 

2 

2 

2 

2 

2 

2 

S 

10 
3 


11 
3 


12 
3 


m: 

9 

12 

5 

8 

11 

4 

7 

10 

13 

4 

4 

4 

4 

4 

4 

4 

4 

4 

Ml 

12 

6 

10 

14 

-   7 

11 

5 

9 

13 

5 

5 

5 

5 

5 

5 

5 

5 

5 

Mj 

10 

14 

9 

13 

8 

12 

7 

1.1 

6 

_6 

6 

6 

6 

6 

6 

6 

6 

6 

M!l 

16 

9 

12 

15 

8 

11 

14 

7 

10 

7 

7 

7 

7 

7 

7 

7 

7 

_7 

R<12 

16 

10 

14 

8 

11 

15 

13 

9 

17 

~^A 

8 

8 

8 

8 

8 

8 

8 

8 

_8 

M^ 

14 

17 

10 

13 

16 

12 

9 

18 

15 

_9 

9 

9 

_9 

9 

9 

9 

9 

9 

Note. — The  preceding  exercises  are  to  be  written  upon  the  board,  and 
used  in  class  drill  daily,  until  every  pupil  can  give,  at  sight,  the  difference 
of  any  two  of  the  numbers. 

Oral  Work.  —  ^5.  A.  From  every  tenth  nnmber, 
from  18  to  98  inclusive, 

1.  Subtract  1. 

2.  Subtract  2. 


3.  Subtract  3. 
Jf.  Subtract  4. 
5.  Subtract  5. 


6.  Subtract  6. 

7.  Subtract  7. 

8.  Subtract  8. 


9.  Subtract    9. 
10.  Subtract  10. 


38 


SECOND    BOOK  IN  ARITHMETIC. 


S.  From  every  tenth  number,  from  19  to  99  inclusive, 


1.  Subtract  1. 
^.  Subtract  2. 


3.  Subtract  3. 
4"  Subtract  4. 
5.  Subtract  5. 


6.  Subtract  6. 

7.  Subtract  7. 

8.  Subtract  8. 


9.  Subtract    9. 
10.  Subtract  10. 


C  From  every  tenth  number,  from  10  to  100  inclusive, 


1.  Subtract  1. 

2.  Subtract  2. 


3.  Subtract  3. 
4-.  Subtract  4. 
5.  Subtract  5. 


6.  Subtract  6. 

7.  Subtract  7. 

8.  Subtract  8. 


9.  Subtract    9. 
10.  Subtract  10. 


56.  1.  From  9  take  7 ;  take  5 ;  take  2. 
^.  From  14  take  6 ;  take  10 ;  take  5. 
From  18  take  10 ;  take  8  ;  take  0. 


S. 

6. 
6. 


take  7 ;  take  11. 
take  9  ;  take  6. 
from  15  ;  from  12. 
from  7 ;  from  10. 
from  17 ;  from  13. 
from  15  ;  from  13. 
from  7 ;  from  14. 


From  11  take  3 
From  16  take  7 
Take  6  from  11 

7.  Take  3  from  12 

8.  Take  8  from  15 

9.  Take  9  from  19 

10.  Take  7  from  15 
How  many  are 

11.  32-25?     20-13?    47-6?     21-9?     30-23? 

1^.  17  minus  9  ?     25  minus  18  ?     34- 7 ?     15- 6  ? 

13.  28  less  18?     45  less  8?     16  less  11?    43  less  35? 
50  less  41  ? 

Find  the  value  of  each  of  these  expressions  : 
U.  Apples:  21-7;  18-10;  26-8;  32-20. 

15.  Oranges :  38  less  29 ;  62  less  55 ;  48  less  7;  24  less  10. 

16.  Lemons  :  52  minus  40  ;  31  minus  27 ;  23  minus  9  ; 
16  minus  7. 

17.  Figs :  14  from  23 ;  9  from  30 ;  12  from  18 ;  48  from  56. 

18.  Peaches :  15  from  31 ;  12  from  29  ;  20  from  45. 


INTEGERS,— 8  UB  TRA  (JTION.  39 

^1.  A.  (a)  75  pears,  36  dollars,  115,  283  dollars,  491 
pears. 

1.  What  number  in  line  (a)  can  be  subtracted  from 
283  dollars?     Why? 

^.  What  number  from  491  pears  ?  Why  ?  What  num- 
ber from  115  ?     Why  ? 

S.  Ones  can  be  subtracted  from  what  only  ? 

^.  Tens  must  be  subtracted  from  what  ?  Hundreds, 
from  what  ? 

5.  Subtract  10  from  50 ;  100  from  500 ;  1,000  from 
5,000  ;  10,000  from  50,000. 

6.  Take  $  .04  from  $  .09  ;  $  .40  from  $  .90  ;  $40  from 
$90  ;  $400  from  $900 ;  $4,000  from  $9,000. 

7.  From  10  subtract  7 ;  from  100  subtract  70  ;  from 
1,000  subtract  700  ;  from  10,000  subtract  7,000. 

8.  From  $  .20  subtract  $  .04 ;  from  $2  subtract  $  .40. 

9.  From  $20  subtract  $4 ;  from  $200  subtract  $40. 

-B.  1,  30  is  2  tens  and  how  many  ones  ? 
^.  37  is  2  tens  and  how  many  ones  ? 
3.  54  is  40  and  how  many  ? 
4"  300  is  2  hundreds  and  how  many  tens  ? 

5.  370  is  2  hundreds  and  how  many  tens  ? 

6.  268  is  25  tens  and  how  many  ones  ? 

7.,  1,000  is  9  hundreds,  9  tens,  and  how  many  ones? 

8.  10,000  is  0  ones,  10  ten^,  and  how  many  hundreds 
and  thousands  ? 

9.  420  is  300  and  how  many  ? 

10.  423  is  410  and  how  many  ? 

11.  900  is  800  and  how  many  tens  ? 

12.  905  is  890  and  how  many  ? 


40 


SECOND   BOOK  IN  ARITHMETIC. 


Process. 

^0^3  75  Minaeud. 

^,869  Subtrahend. 

3  7.506  Difference. 


58.  Principle  of  Subtraction. 
Units  of  any  order  can  he  svhtracted  from  units  of  the 
same  order  only. 

Written  Wovh. — 59,  Ex.  What  is  the  difference  be- 
tween 40,375  and  2,869  % 

Explanation. — I  write  the  subtra- 
hend under  the  minuend  —  ones 
under  ones,  tens  under  tens,  and 
so  on — and  draw  a  horizontal  line 
below  the  subtrahend. 

Since  9  ones  can  not  be  taken  from 
5  ones,  I  take  from  the  7  tens  1  ten 

(=  10  ones),  and  unite  it  with  the  5  ones,  making  15  ones ;  then, 
9  ones  from  15  ones  leave  6  ones,  which  I  write  in  the  result. 

6  tens  from  the  remaining  6  tens  of  the  minuend  leave  0,  which  I 
write  in  tens'  place  in  the  result. 

Since  8  hundreds  can  not  be  taken  from  3  hundreds,  and  there 
are  0  thousands  in  the  minuend,  I  take  from  the  4  ten-thousands 
1  ten-thousand  (=10  thousands);  from  these  10  thousands  I  take 
1  thousand  (=  10  hundreds),  and  unite  it  with  the  3  hundreds, 
making  13  hundreds ;  then,  8  hundreds  from  13  hundreds  leave 
5  hundreds,  which  I  write  in  the  result. 

2  thousands  from  the  remaining  9  thousands  leave  7  thousands, 
which  I  write  in  the  result. 

Since  there  are  0  ten-thousands  in  the  subtrahend,  I  write  the  re- 
maining 3  ten-thousands  of  the  minuend  in  the  result. 

The  entire  result,  37,506,  is  the  required  difference. 

Problems. 

JL       1  2  3  4  5  6 

584      725     680      805      943      1,308 
152      314     244      431      576      1,245 


635  pins 
412  pins 

M 

$57,698 
43,257 


8 

3,544  soldiers 
2,836  soldiers 

12 

$6,750.02 
2,945.04 

9 

7,953  shingles 
5,468  shingles 

13 

$42,301.00 
5,174.56 

59,700  pounds. 
43,708  pounds 

14- 
$16,065.00 
10,438.25 


INTEGERS,— 8UBTRA  CTION. 


41 


Oral  Work, — Exercises  to  be  solved  at  sight 

B 

From 

Take 

From 

Take 

1. 

2. 

13. 

u. 

S. 

From  87 

50 

15. 

From  25  cents 

15  cents 

-^. 

Take   60 

20 

16. 

Take  13  cents 

6  cents 

5. 

6. 

17. 

18. 

7. 

From  99 

73 

19. 

From  65  cents 

50  cents 

8. 

Take   62 

41 

20. 

Take  37  cents 

25  cents 

9. 

10. 

21. 

22. 

11. 

From  78 

48 

23. 

From  100  cents 

75  cents 

12, 

Take  57 

43 

21 

Take     87  cents 

31  cents 

€ 

Minuends. 

Subtrahends.               Minuends. 

Subtrahends. 

1. 

2. 

5. 

6. 

3. 

Minuends 

44  hats 

34  hats 

7.  57  coats 

43  coats 

4- 

Subtrahends 

16  hats 

9  hats 

^.36  coats 

19  coats 

9. 

10. 

13. 

U- 

11. 

Minuends 

84  tons 

69  tons 

15.  64  pens 

55  pens 

12. 

Subtrahends  78  tons 

45  tons 

16.  39  pens 

24  pens 

17. 

18. 

21. 

22. 

19. 

Minuends 

$2.00 

$.50 

23.  $10.00 

$5.00 

20. 

Subtrahends  $  .75 

$.37 

2Jf.  $  2.50 

$  .50 

jD.  1.  A  man  bought  a  cow  for  $30,  and  paid  all  but 

How  much  did  he  pay  ? 
He  paid  the  difference  between  $30  and  $10,  which  is  $20. 

2.  Julia  had  100  cents,  but  she  paid  20  cents  for  some 
paper.     How  many  cents  had  she  left  ? 

S.  60  gallons  of  syrup  have  been  drawn  from  a  hogshead 
that  contained  80  gallons.     How  many  gallons  are  left  ? 

^.  A  fisherman  caught  110  pounds  of  fish,  and  sold  70 
pounds.     How  many  pounds  had  he  left  ? 


42  SECOND    BOOK   IN   ARITHMETIC. 

5.  A  man  having  $63  paid  $20  for  a  harness.     How 
many  dollars  had  he  left  ? 

6.  Of  the  75  pupils  in  a  school  40  are  girls.     How 
many  pupils  are  boys  \ 

7.  A  girl  having  79  chickens,  sold  30  of  them.     How 
many  chickens  had  she  left  ? 

8.  A  gentleman  paid  $87.  for  a  sleigh,  and  $60  less  for 
a  robe.     How  much  did  the  robe  cost  him  ? 

9.  A  paper-box  maker  having  72  sheets  of  straw-board, 
used  60  of  them.     How  many  sheets  had  he  left  ? 

10.  I  buy  potatoes  at  90  cents  a  busliel,  and  sell  them 
at  108  cents.     How  much  do  I  gain  on  a  bushel  ? 

Written  Work. 

B      1                    1  i  i 

348,794    2,174,943  84,125,8(5o  167,065,149 

127,586      480,765  9,632,486  39,999 

5  6  7  8  9 

104,021  $9,004  50,096  $123.45  1,000,000 

99,034  2,876  13,188  109.86  31,276 

ion  1?  !^ 

7,408,215  300,300,333  $18,123.05  $386.00 

59,326  47,008,296  3,109.86  .21 

$  $ 

Oral  Work. — F.  1.  A  merchant  tailor  used  15  yards 
of  broadcloth  from  a  piece  containing  39  yards.  How 
many  yards  were  left  in  the  piece  ? 

There  was  left  the  difference  between  39  yards  and  15  yards. 
10  yards  from  39  yards  leave  29  yards,  and  5  yards  from  29 
yards  leave  24  yards. 

%.  A  shoe  dealer  having  48  pairs  of  ladies'  shoes,  sold 
25  pairs.     How  many  pairs  had  he  left  ? 


INTEGERS.— 8UBTRA  CTION.  4B 

3.  I  own  a  farm  of  69  acres,  and  58  acres  of  it  are 
cleared  land.     How  many  acres  are  woodland  ? 

^.  A  cooper  made  53  apple  barrels  in  a  week,  and  sold 
44  of  them.     How  many  barrels  had  he  left  ? 

5.  A  fruit  dealer  bought  72  crates  of  peaches.  After 
Belling  56  crates,  how  many  crates  had  he  yet  to  sell  ? 

6.  A  wood  dealer  sold  83  cords  of  oak  wood,  and  57 
cords  of  pine.  How  many  more  cords  of  oak  than  pine 
wood  did  he  sell  ? 

7.  A  butcher  killed  one  calf  that  weighed  91  pounds, 
and  another  that  weighed  26  pounds  less.  How  much  did 
the  lighter  calf  weigh  ? 

8.  A  man  owing  $75,  gave  his  note  for  $47,  and  paid 
the  balance  in  cash.     How  much  cash  did  he  pay  ? 

9.  I  exchanged  a  horse  worth  $84,  for  a  gold  watch 
and  $19  in  money.     How  much  did  the  watch  cost  me  ? 

10.  111  swallows  are  how  many  more  than  28  swallows? 

11.  108  robins  —  99  robins  are  how  many  robins  ? 

12.  56  cents  are  how  many  less  than  100  cents? 

Written  Work. — G.  1.  How  many  tons  are  52,719 
tons --24,112  tons? 

2.  A  fruit  dealer,  having  1,247  baskets  of  peaches,  sold 
965  baskets.     How  many  baskets  had  he  left  ? 

3.  A  man  whose  income  is  $2,250  a  year,  expends 
$1,473.     How  much  does  he  save  ? 

Jf..  A  dealer  bought  horses  for  $4,960,  and  sold  them 
for  $6,424.     How  much  did  he  gain  ? 

5.  In  a  certain  city  are  16,447  children,  of  whom  11,096 
attend  school.     How  many  do  not  attend  school  ? 

6.  A  man  having  $17,974  in  the  bank,  drew  out  $8,598. 
How  much  money  had  he  left  on  deposit  ? 


44  SECOND    BOOK  IN  ARITHMETIC. 

7,  In  November  a  mercliant's  sales  amounted  to 
$2,928.93,  and  in  December  to  $3,743.69.  How  much 
less  were  the  November  sales  than  those  of  December  ? 

8,  A  gentleman's  income  last  year  was  fifteen  thou- 
sand four  hundred  thirty-one  dollars,  and  his  expenses 
were  nine  thousand  three  hundred  fifty  dollars  nineteen 
cents.     How  much  did  he  save  ? 

9,  One  year  the  value  of  my  farm  products  was 
$4,307,  and  my  farm  expenses  were  $2,427.  How  much 
were  my  profits  ? 

10.  A  ship  builder  received  $19,000  for  a  brig  that  cost 
him  $16,728.     How  much  was  his  gain  ? 

11.  A  miller  in  St.  Louis  has  36,500  barrels  of  fiour. 
How  many  barrels  will  he  have  in  store,  after  shipping 
21,987  barrels  to  New  Orleans  ? 

W.  A  broker  sold  Government  bonds  for  $328,700  that 
cost  him  $281,908.     How  much  did  he  gain  ? 

13.  How  many  acres  are  1,367  acres  —  1,108  acres? 
IJi,.  How  many  cords  are  22,197  cords  —  20,174  cords  ? 

15.  103,035  feet  -  83,616  feet  r:.  how  many  feet  ? 

16.  A  load  of  hay  and  the  wagon  weighed  2,481  pounds, 
and  the  wagon  weighed  812  pounds.  What  was  the  weight 
of  the  hay  ? 

17.  At  a  city  election  one  candidate  for  mayor  received 
52,918  votes,  and  the  other  candidate  28,434  votes.  What 
was  the  majority  of  the  successful  candidate  ? 

18.  The  Island  of  Cuba  contains  43,220  square  miles, 
and  the  State  of  Ohio  41,060  square  miles.  How  much 
larger  is  Cuba  than  Ohio  ? 

19.  The  live  weight  of  ten  beeves  was  9,742  pounds, 
and  the  dressed  weight  was  6,495  pounds.  How  much 
was  the  shrinkao:e  ? 


INTEGERS.— 8UBTRA  CTION. 


45 


60.  Etjle  for  Subteaction  of  Integees. 

I.  Write  the  subtrahend  under  the  minuend — ones  un- 
der ones^  tens  under  tens^  and  so  on. 

II.  Beginning  at  the  right ^  subtract  the  units  of  each 
order  in  the  subtrahend  from  the  units  of  the  same  order 
in  the  minuend^  and  write  the  difference  in  the  result. 

III.  When  there  are  more  units  of  any  order  in  the 
subtrahend  than  of  the  same  order  in  the  minuend^  add 
Vd  to  the  units  of  the  minuend  before  subtracting y  then^ 
consider  the  units  of  the  next  higher  order  in  the  minu- 
end 1  less. 

Problems. 


A.     From 

Take 

From                 Take 

1.            9,426 

8,563. 

^. 

9,111,525,202           527,029 

2.      $400,385 

$27,690. 

5. 

$6,000.07           $375.68 

S.  69,504,300 

609,572. 

6. 

$22,025.15       $19,936.93 

B.  Ex.  Subtract  2,869 

Making  the  Computation. 

from  40,375 

9  from  15,  6;  write  6. 

Process. 

6  from    6,  0;  write  0. 

U0,S75 

8  from  13,  5;  write  5. 
2  from    9,  7;  write  7. 

2,869 

Write  3. 

37,506  Result,  37,506. 

Subtract  eacli  number  in  column  I  from  the  numbers 
on  the  same  line  in  columns  II,  III,  IV ;  the  numbers  in 
column  II  from  those  in  columns  III  and  IV ;  the  num- 
bers in  column  III  from  those  in  column  IV. 


i-  6. 

7-12. 
13-18. 
19-24. 


I. 

77 
160 

80 

1.75 


II. 

463 

1,098 

515 


25-30.  $.05 


$8.91 


III. 
645 
2,150 
9,100 

$4,759 
$64.07 


IV. 

48,100 

10,360 

100,125 

$98,467 
$180.63 


46  SECOND    BOOK  IN  ARITHMETIC. 

SI,  What  is  the  difference  between  nine  thousand  nine- 
teen and  seven  thousand  twenty-one  ? 

32,  From  fifteen  million  two  thousand  four  take  eighteen. 

33,  From  100  thousand  82  take  1  thousand  9. 

3^.  The  greater  of  two  numbers  is  11,419,  and  the  less 
is  7,255.     What  is  the  difference  ? 

Oral  Work. — C.  1,  A  merchant  pays  one  clerk  $35  a 
month,  and  another  $9  less.  How  much  are  the  monthly 
wages  of  the  second  clerk  ? 

2,  From  a  board  fifteen  feet  long,  a  carpenter  cut  off  a 
piece  nine  feet  long.  What  was  the  length  of  the  part 
left? 

3,  A  gentleman  paid  $22  for  a  lawn  mower,  and  after 
using  it  one  season,  sold  it  for  $9  less  than  cost.  How 
much  did  he  receive  for  it  ? 

^.  .If  you  start  from  New  York  and  travel  north  90 
miles,  and  then  south  50  miles,  how  far  will  you  be  from 
New  York? 

6.  An  ice  harvester  housed  97  tons  of  ice  one  week, 
and  27  tons  less  the  following  week.  How  many  tons  did 
he  house  the  second  week  ? 

6,  Mark  had  60  cents,  and  Thomas  had  46.  Mark 
spent  as  many  cents  as  Thomas  earned,  and  he  then  had 
51  cents.     How  many  cents  did  Thomas  then  have  ? 

7,  Peter  husked  35  bushels  of  corn  Monday,  and  62 
bushels  Monday  and  Tuesday.  How  many  more  bushels 
did  he  husk  Monday  than  Tuesday  ? 

Written  Work. — D.  1.  Oliver  Goldsmith  died  in  the 
year  1774,  at  the  age  of  46.     In  what  year  was  he  born  ? 

2.  A  man  died  in  the  year  1799,  aged  67  years.  In 
what  year  was  he  born  ? 


INTEGERS.— SUBTRA  CTION.  47 

How  many  years  elapsed 

S.  From  the  settlement  of  St.  Augustine  in  1565,  to 
tlie  settlement  of  :N"ew  York  in  1613  ?  " 

^.  Between  the  settlement  of  St.  Augustine,  and  the 
settlement  of  Plymouth  in  1620  ? 

5.  From  the  settlement  of  St.  Augustine  to  the  Dec- 
laration of  Independence  in  1776  ? 

6.  From  the  settlement  of  Plymouth  until  the  discov- 
ery of  gold  in  California  in  1818  ? 

7.  From  the  discovery  of  America  to  the  present  year  ? 

8.  The  taxes  on  a  town  are  $11,000 ;  of  this  amount 
$5,680  are  county  and  state  tax.     What  is  the  town  tax  ? 

9.  The  receipts  of  a  machine-shop  for  a  year  are 
$33,296,  and  the  running  expenses  are  $22,535.  What 
are  the  net  earnings  for  the  year  % 

10.  Mount  Sorata,  a  peak  of  the  Andes,  is  21,286  feet 
high,  and  5,506  feet  higher  than  Mont  Blanc,  the  highest 
peak  of  the  Alps.     How  high  is  Mont  Blanc  ? 

11.  A  contractor  delivered  50,000  rifles  for  the  army, 
but  only  41,715  of  them  were  accepted.  How  many  of 
them  were  condemned  as  imperfect  ? 

12.  The  minuend  is  one  hundred  twenty-five  million 
three  hundred  sixty -five  thousand  nine  hundred  forty- 
eight,  and  the  subtrahend  is  eight  million  seven  hundred 
thousand  nine  hundred  sixty-six.    What  is  the  remainder  ? 

IS.  26,957,229  bushels  of  salt,  less  11,094,227  bushels 
are  how  many  bushels  ? 

U.  A  is  worth  $25,864;  B  is  worth  $1,350  less  than 
A ;  C  $965  less  than  B  ;  and  D  $7,393  less  than  C.  How 
much  is  D  worth  % 

Note.— For  outlines  of  subtraction  for  review,  see  page  269. 


48  SECOND   BOOK  IK  ARITHMETIC. 

Review  Problems. 

Oral  Work L  From  40  +  8  +  16  +  9  +  7  +  8  +  11 

take  20  +  10  +  25  +  7. 

2,  From  13  +  10  +  5  +  12  +  7+9  take  ^^Q-^-Q^^^, 

3.  From  250  take  75  +  10  +  25  +  50. 

^.  From  175-45  take  25  +  25  +  9  +  9. 

6.  A  boj  who  had  45  doves,  sold  20,  and  afterward 
bouglit  17  more.     How  many  doves  had  he  then  ? 

6,  A  bell  maker  melted  together  25  pounds  of  tin  and 
6  pounds  of  copper.  He  used  4  pounds  of  the  bell-metal 
for  call-bells,  8  pounds  for  hand-bells,  and  the  remainder  for 
sleigh-bells.    How  many  pounds  did  he  use  for  sleigh-bells  ? 

'^.  A  man  owing  $120,  paid  $49  one  day,  and  $28  the 
next  day.     How  much  did  he  then  owe  ? 

8.  Luther  had  $  .23,  and  earned  $  .44.  He  then  spent 
$  .27,  and  lost  $  .05.     How  many  cents  had  he  left  ? 

9.  Jasper  has  15  peaches  in  a  basket,  4  in  his  pockets, 
and  3  in  his  hand.  If  he  gives  5  peaches  to  Ann,  and  7 
to  Jane,  how  many  peaches  will  he  then  have  ? 

10.  A  boy  gave  a  dollar  bill  to  pay  for  a  slate  that  cost 
$  .36,  a  writing-book  that  cost  $  .20,  and  some  ink  that  cost 
$  .15.     How  much  change  should  he  receive  ? 

Written  Work. — 1.  A  father  and  his  two  sons  earned 
$2,395  in  a  year,  the  elder  son  earning  $709,  and  the 
younger  son  $531.     How  much  did  the  father  earn  ? 

2.  On  a  certain  day  93,825  persons  entered  the  city  of 
New  York.  Of  this  number  26,759  came  by  railroads, 
60,048  by  steamboats,  and  the  others  by  steamships  and 
sailing  vessels  from  foreign  countries.  How  many  arrived 
from  foreign  countries  ? 


INTEGERS.—SUBTRA  CTION.  49 

5,  A  dealer  paid  $25,267  for  horses,  his  expenses  in 
taking  them  to  market  were  $6,485,  and  he  sold  them  for 
$37,496.     How  much  were  his  profits  ? 

^.  From  the  sum  of  thirty-six  million  five,  one  hun- 
dred five  thousand  seven  hundred  one,  and  nine  million 
nine  thousand  ninety,  subtract  the  sum  of  ninety-six  thou- 
sand three  hundred,  and  forty-two  thousand  nine. 

6,  A  woman  received  $439  for  early  vegetables,  and 
$218  for  poultry.  The  expense  of  raising  the  vegetables 
was  $124,  and  of  the  poultry  $84.    What  were  her  profits  ? 

6.  A  speculator  at  one  time  gained  $6,760,  and  then 
lost  $3,400 ;  at  another  time  he  gained  $3,650,  and  then 
lost  $3,954.     How  much  did  he  gain  in  all? 

7.  January  1,  A  had  property  worth  $10^350.75,  but 
he  owed  $1,050.31.  During  the  year  he  earnsd  $1,156, 
and  expended  $975.25,  and  his  property  increased  $550. 
How  much  was  he  worth  at  the  end  of  the  year  \ 

8.  A  grocer  purchased  370  hogsheads  of  sugar  weigh- 
ing 372,960  pounds,  and  sold  195  hogsheads  weighing 
196,725  pounds.  How  many  hogsheads  had  he  left,  and 
what  was  their  weight  ? 

9.  In  ^YQ  weeks  415,678  tons  of  coal  were  carried  to 
Philadelphia  by  the  Eeading  Railroad,  234,509  tons  of 
which  were  carried  in  the  first  four  weeks.  How  many 
tons  were  carried  the  fifth  week  ? 

10.  On  the  first  of  January  an  edition  of  11,000  copies 
of  a  book  was  published.  In  January  996  copies  were 
sold,  in  February  1,025,  in  March  2,363,  in  April  1,808,  in 
May  845,  and  in  June  2,471.  How  many  copies  remained 
unsold,  July  1  ? 

Note.— Teachers  who  prefer  that  decimals  should  be  studied  in  connec 
tion  with  integers,  will  now  require  their  pupils  to  study  subtraction  oJ 
decimals,  pages  123-125. 

0 


SECTION  IV. 

MULTIPLICATION. 

Oral  Work.  —  61.  A.  1,  A  wagon  has  4  wheels. 
How  many  wheels  have  3  wagons? 

Three  -wagons  have  the  sum  of  4  -wheels,  and  4  -wheels,  and 

4  wheels,  vrhich  is  12  -wheels.     Or, 

Three  -wagons  have  3  times  4  -wheels,  which  are  12  -wheels. 

2,  How  many  wlieels  have  4  wagons  ? 

S,  How  many  feet  have  5  horses  ? 

.^.  How  many  hands  have  2  men  ?     3  men  ?     4  men  ? 

5,  How  many  cherries  are  there  in  2  clusters  of  3 
cherries  each  ?     In  3  clusters  ?     In  4  clusters  ? 

^.  How  many  cents  are  3  5-cent  pieces  ?    Are  5  5-cent 
pieces  ? 
How  many  are 

7.  3  cherries  and  3  cherries,  or  2  times  3  cherries  ? 

8,  4  boxes  and  4  boxes,  or  2  times  4  boxes  ? 

P.  3  hats,  and  3  hats,  and  3  hats,  and  3  hats,  or  4  times 
3  hats? 

10.  4  mats,  and  4  mats,  and  4  mats,  or  3  times  4  mats  ? 
IL  %  .02,  and  $  .02,  and  $  .02,  and  $  .02,  and  $  .02,  or 

5  times  2  cents? 

1^.  5  dimes  and  5  dimes,  or  2  times  5  dimes  ? 
IS.  5  times  $3,  or  3  times  $5  ?     5  times  4  men,  or  4 
times  5  men? 

JK.  i.  How  many  are  2  +  2?     2  +  2  +  2?     2  +  2  +  2  +  2? 
2.  How  many  are  two  2's  ?     Three  2's  ?     Four  2's  ? 
S.  How  many  are  2  times  2  ?     3  times  2  ?     4  times  2  ? 
Add  Subtract 

Jf..  By  2's,  from  0  to  20.  5.  By  2's,  from  20  to  0. 

6.  By  3's,  from  0  to  30.  7.  By  3's,  from  30  to  0. 


INTEGERS.-^MULTIPLICA  TION.  51 

05.  Multiplication  is  the  process  of  finding  the 
sum  of  one  of  two  numbers  taken  as  many  times  as  tliere 
are  ones  in  the  other. 

a»  The  number  to  be  taken  is  the  multiplicand, 

b.  The  number  that  shows  how  many  times  the  multipHcand 
is  to  be  taken  is  the  multiplier. 

c.  The  multiplicand  and  multiplier  are  factors, 

d.  The  number  obtained  by  multiplying  is  the  product. 
Multiplication  may  also  be  defined — a  short  process  of  add- 
ing equal  numbers. 

Exercises. 
A.  1  times  8  are  56. 
'    1,  Which  of  these  numbers  is  the  multiplicand? 

^.  Which  is  the  multiplier  ?     Which  is  the  product  ? 

3.  Which  are  the  factors  ? 

Jf.  What  is  the  product  of  the  factors  10  and  Y  ? 

6,  What  is  the  product  of  6  times  8  loaves  of  bread  ? 

6.  The  factors  are  5,  4,  and  7.     What  is  the  product  ? 

7,  The  multiplicand  is  8,  and  the  multiplier  is  5.  What 
is  the  product  ? 

IB.  In  a  garden  are  5  rows  of  fruit-trees,  and  in  each 
row  are  7  trees.     How  many  trees  are  in  the  garden  ? 

First  Solution. — In  the  garden  are  7  trees  +  7  trees  -|-  7  trees  -j- 
7  trees,  -f-  7  trees,  which  are  35  trees.     Or, 

Second  Solution. — ^In  the  garden  are  5  times  7  trees,  "which  are 
35  trees. 

7  trees,  the  multiplicand  in  the  second  solution,  is  one  of  the  equal 

parts  in  the  first  solution. 
5,  the  multiplier  in  the  second  solution,  is  the  number  of  equal 

parts  in  the  first  solution. 
35  trees,  the  product  in  the  second  solution,  is  th^  sum  in  the  first 

solution. 


52 


SECOND    BOOK  IN  ARITHMETIC. 


63.  Add 

1.  By  4's,  from  0  to  40. 
3.  By  5's,  from  0  to  3€>. 
5.  By  6's,  from  0  to  60. 


Subtract 

2,  By  4's,  from  40  to  0. 
Jf.  By  5's,  from -50  to  0. 
6,  By  6's,  from  60  to  0. 


64.  The   sign  of  multiplication   is   an   oblique 
cross,  X-     I^  is  ^^^^  ti7nes^  or  midtijplied  hy, 
5x8  may  be  read  **  5  times  8,"  or  "  5  multiplied  by  8." 

A.  Read  these  exercises : 


1,  '7x12=   84 
^.  4x25  =  100 

7.  8x5x6  =  10x9 

8.  2X6X12  =  3X8X6 


3.       9x63=    567 
Jf.   144x25  =  3,600 


10. 


5.  3X8  =  4X    6 

6.  3x4x5  =  60 
16x14  =  4x7x8 
6X9X4=2X3X6X6 


Written  Work. 

B.  Write  each  of  these  exercises,  using  the  proper  signs : 

Jf.  $2.25  multiplied  by  5  are  $11.25. 

5.  The  product  of  2,  3,  4,  and  5  is 
120. 

6.  4  times  $  .63  are  $2.52. 


1.  5  times  25  are  125. 

2.  15  times  4  quarts 

are  60  quarts. 

3.  3  times  7  equal  21. 

C  Copy  and  complete 

1.  4X    9  = 

2.  9x   4  = 

5,  8x    7  = 

^.      5X10X2  = 

6.  6x7  melons  =     melons. 
^.  lOx   4  apples  =     apples. 

Oral  Work. 
Z>.  Add 

1.  By  7's,  from  0  to  70. 
3.  By  8's,  from  0  to  80. 
5.  By  9's,  from  0  to  90. 


7.  4applesxl0=    apples. 

8.  3x2x5  balls  =    balls. 

9.  4x$20  =  $ 

10.  8X$.05=$ 

11.  $1.50X2=$ 

12.  $5.10X3  =  $ 


Subtract 

2.  By  7's,  from  70  to  0. 
Jf.  By  8's,  from  80  to  0. 
6.  By  9's,  from  90  to  0. 


Written  Work.-^S.  Copy,  complete,  learn,  and  recite  the 


INTEGERS.— MUL  TIPLICA  TION, 


53 


Table  of  Peimaky  Combinations  in  Multiplication. 


1X1  = 

2X2  = 

3X3  = 

4x4  = 

2X1  = 

3X2  = 

4X3  = 

5X4  = 

3x1  = 

4x2  = 

5X3  = 

6X4  = 

4X1  = 

5X2  = 

6X3  = 

7x4  = 

5X1  = 

6X2  = 

7x3  = 

8X4  = 

6X1  = 

7x2  = 

8X3  = 

9X4  = 

7x1  = 

8X1  = 
9X1  = 

8X2  = 
9X2  = 

7x7  = 

9X3  = 

Qx^= 

7x6  = 

5X5  = 
6X5  = 
7x5  = 

8X8  = 

8x7  = 

8X6  = 

8X5  = 

9X9  = 

9x7  = 

9X6  = 

9X5  = 

9x9  = 

06.  Exercises  in  Multiplication  at 

Sight. 

.    S       2 

0         8         3 

6         4          5 

7        1 

5 

3 

2 

0 

7 

9 

8 

1 

6 

4 
3 

1 

9 

7 

8 

5 

0 

4 

6 

3 

2 
4 

9 

2 

7 

1 

6 

5 

4 

0 

8 

3 
5 

4 

6 

2 

9 

1 

8 

5 

3 

0 

7 
6 

8 

5 

6 

2 

0 

7 

3 

9 

1 

4 

7 

3 

9 

1 

6 

2 

5 

0 

8 

4 

7 
8 

6 

1 

0 

5 

3 

9 

2 

4 

8 

7 
9 

Note. — The  preceding  exercises  are  to  be  written  upon  the  board,  and 
used  in  class  drill  daily,  until  every  pupil  can  give,  at  sight,  the  product  of 
the  numbers  expressed  by  any  two  digits. 


54:  SECOND    BOOK  IN   ARITHMETIC. 

67.  A  number  is  either  concrete  or  abstract. 

68.  A  concrete  number  is  a  number  whose  unit  is 
named. 

69.  An  abstract  number  is  a  number  whose  unit 
is  not  named. 

a.  5  apples,  70  days,  $658  are  concrete  numbers. 

b.  One,  sixty-five,  19,  432  are  abstract  numbers. 

a.  Three,  seven,  four  books,  nine  men. 

b.  13  boys,  50,  72  hours,  365  days,  10,243. 

1,  In  line  a  which  numbers  are  abstract  ?    Which  are 
concrete  '^     Why  ? 

^.  In  line  b  which  numbers  are  concrete?    Which  are 

abstract  ?     Why  ? 

3.  What  is  the  unit  of  each  number  in  line  a  ?     In  line 
b  ?    Is  it  abstract  or  concrete  ? 

Exercises. 

Oral  Work, — 1.  Five  weeks  are  how  many  days  ? 

£,  How  many  hills  of  corn  are  there  in  8  rows  of  10 
hills  each  ?     In  5  rows  of  7  hills  each  ? 

S.  5X    8=: How  many? 
4.  3x50=:how  many  ? 

How  much  must  be  paid  for 


5.  Multiply  15  by      4. 

6,  Multiply    9  by  100. 


7.  8  oranges,  at  5  cents  apiece  ? 

8,  4  readers,  at  $.50  apiece? 


9.  2  coats,  at  15  dollars  each? 
10.  100  sheep,  at  $3  a  head? 


11.  In  each  of  these  ten  exercises,  which  number  is  the 
multiplicand  ?     Is  it  an  abstract  or  a  concrete  number  ? 

12.  In  which  of  these  exercises  is  the  product  an  ab- 
stract number? 

13.  In  which  exercises  is  it  a  concrete  number  ? 


INTEGERS.— MUL  TIPLICA  TIOK 


55 


In  each  of  the  ten  exercises, 
H.  Which  number  is  the  multiplier? 
15.  Is  it  abstract  or  concrete? 

The  multiplie?'  is  always  considered  an  abstract  number. 

What  is  the  product 


16.  Of  4  times  3  ? 

17.  Of  3  times  4  ? 


18.  Of  5  times  9  ? 

19.  Of  9  times  5  ? 


21. 


Of  15x50? 
Of  50  X  15  ? 


In  written  work,  either  factor  may  be  used  as  the  multiplier. 

70.  Peinciple  I.  The  multijplicand  and  jproduct  are 
lihe  numbers. 

Case  I.   The  multiplier  a  digit. 


71.  1.  Multiply 

20  by  4. 
4  times  2  tens  or 

20  are  8  tens  or 

80. 

Multiply 

i.  30,   50,   70,  and  40  by  4. 

5.  13,  24,  51,  and  32  by  5. 

6.  17,  47,  77,  and  87  by  7. 

7.  25,  75,  19,  and  69  by  9. 


2.  Multiply  32 
by  3. 

3  times  2  are  6,  3 
times  30  are  90, 
and  90+6  are  96. 


3.  Multiply  53  by  5. 

5  times  3  are  15,  5 
times  50  are  250, 
and  250  +  15  are 
265. 


8.  $36  by  7,  4,  3,  and  9. 

9.  $  .25  by  8,   5,  7,  and  3. 

10.  48  pounds  by  3,  6,  and  8. 

11.  54  yards  by  4,  7,  and  9. 


7^.  A.  1.  If  a  man  can  cut  8  acres  of  grass  with  a 
reaper  in  one  day,  how  many  acres  can  he  cut  in  3  days  ? 

In  3  days  he  can  cut  3  times  as  many  acres  as  in  .one  day, 
and  3  times  8  acres  are  24  acres. 

2.  How  much  will  4  saddles  cost,  at  $9  apiece  ? 

3.  A  shipper  of  Western  beef  ships  4  car  loads  a  day. 
How  many  car  loads  does  he  ship  in  7  days  ? 

Jp.  If  8  bushels  of  apples  will  make  a  barrel  of  cider, 
how  many  bushels  will  make  9  barrels  ? 

5.  How  much  are  3  acres  of  hay  worth,  at  $20  an  acre  ? 


56  SECOND    BOOK  IN  ARITHMETIC. 

6.  How  many  yards  are  there  in  three  pieces  of  carpet- 
ing of  60  yards  each  ? 

7.  How  many  gallons  of  molasses  are  there  in  5  casks 
of  40  gallons  each  ? 

JB*  1.  How  much  will  4  pounds  of  raisins  cost,  at  22 
cents  a  pound  ? 

^.  How  many  yards  are  there  in  3  pieces  of  linen, 
each  piece  containing  42  yards  ? 

3.  At  $35  apiece,  how  much  will  2  cutters  cost  ? 

^.  If  32  quarts  of  milk  are  used  in  a  hotel  daily,  how 
many  quarts  are  used  in  5  days  ?    In  a  week  ?    In  9  days  ? 

5.  How  many  hours  are  there  in  3  days  ?     In  8  days  ? 

6.  How  much  will  7  bushels  of  potatoes  cost,  at  $  .44 
a  bushel  ?    At  $  .56  ?    At  $  .63  ? 

7.  At  25  bushels  to  the  acre,  how  many  bushels  of  wheat 
can  be  raised  on  5  acres  ?     On  6  acres  ?     On  8  acres  ? 

C.  1,  4  times  2,  plus  4  times  20  equal  4  times  what 
number  ? 

^.  3  times  2,  plus  3  times  40  equal  3  times  what  number  ? 

8.  2  times  $5  +  2  times  $30  =  2  times  how  many  dollars  ? 
4^.  5  times  2  quarts +  5  times  30  quarts  =  5  times  how 

many  quarts  ? 

5.  In  finding  9  times  32  quarts,  what  is  the  first  step  ? 
The  second?     The  third? 

6.  In  multiplying  $16  by  9,  how  many  steps  are  there, 
and  what  are  they  ? 

Give  the  steps,  in  order,  in  multiplying 

7.  24  hours  by  3 ;  by  4 ;  by  7. 

8.  In  finding  7  times  $  .44 ;  7  times  $  .56  ;  7  times  $  .63. 

9.  In  finding  the  product  of  25  bushels  multiplied  by 
5;  by  6;  by  8. 


INTEGERS.— MUL  TIFLICA  TION.  57 

Written  Work.— 7^.  Ex.  Multiply  473  by  9. 

Explanation. —I  write  the  multiplier  under  the  Process. 
ones  of  the  multiplicand,  and  begin  at  the  right  /  7  ? 

to  multiply.  "^ 

9  times  3  ones  are  27  ones,  or  7  ones  and  2  tens.     I ^ 

write  the  7  ones  for  the  ones  of  the  product ; — the  Jf,^^ 5  7 
2  tens  are  a  part  of  the  tens  of  the  product. 

9  times  7  tens  plus  the  2  tens  are  65  tens,  or  5  tens  and  6  hun- 
dreds. I  write  the  5  tens  for  the  tens  of  the  product ; — the  6 
hundreds  are  a  part  of  the  hundreds  of  the  product. 

9  times  4  hundreds  plus  the  6  hundreds  are  42  hundreds,  or  2  hun- 
dreds and  4  thousands,  which  I  write  for  the  hundreds  and 
thousands  of  the  product. 

The  result,  4,257,  is  the  required  product. 


Pkoblems. 

A            1 

Multiply  246 
by       3 

7 

2 

543 

7 

3 

3,269 
5 

8 

4 

9,624 

8 

5 

18,932 
4 

6 

9 

10 

$491              $5,427 
8                        6 

$7,057 
4 

$108.94 
5 

$1,962.40 
9 

II 

20,356  yards 

7 

12                                   13 

257,008  feet        108,094  pounds 
3                            2 

14 

40,007  tons 
9 

B,  1.  A  carriage  maker  sold  7  covered  carriages,  at 
$325  each.     How  much  did  he  receive  for  them  ? 

^.  A  railroad  company  bought  6  locomotives,  at  $12,675 
each.     How  much  did  they  cost  ? 

3.  In  one  week  a  newsboy  sold  246  papers,  at  5  cents 
each.     How  much  did  he  receive  for  them  ? 

Jf.  How  much  will  244  pairs  of  boots  cost,  at  $-4  a  pair  ? 

5.  How  many  pounds  are  there  in  3  barrels  of  salt,  each 
barrel  containing  280  pounds  ? 

C2 


58  SECONB    BOOK   IN   ARITHMETIC. 

6.  A  mile  is  5,280  feet.     8  miles  are  how  many  feet  ? 

7.  What  will  be  the  cost  of  building  a  horse  railroad 
4  miles  long,  at  $12,768  a  mile? 

8.  How  many  gallons  are  8  times  27,645  gallons  ? 

9.  How  many  pounds  are  5  times  32,051  pounds  ? 

10.  What  is  the  product  of  6  times  1,026,348  ? 

11.  How  much  will  a  laborer  earn  in  6   months,  at 
$17.50  a  month  ?     At  $18.75  a  month  ? 

12.  How  much  will  7  tons  of  coal  cost,  at  $6.34  a  ton  ? 

Case  II.   The  multiplier  a  digit  with  a  cipher  or 
ciphers  on  the  right. 

Oral  Work. 


74,  How  much  is 

1.  10  times  5  barrels? 

2.  100  times  5  barrels? 
^.10  times  25  peaches? 
J/..  100  times  25  peaches? 


What  is  the  product 

5.  Of  5  multiplied  by  10? 

6.  Of  5  multiplied  by  100? 

7.  Of  25  multiplied  by  10? 

8.  Of  25  multiplied  by  100? 
Annexing  a  cipher  to  a  number  multiplies  the  number  by  10, 

Hence, 

75.  Principle  II.  Each  removal  of  a  number  one  jpldce 
to  the  left  multijplies  the  number  hy  10. 

1.  At  the  rate  of  4  bushels  an  hour,  how  many  bush- 
els of  potatoes  can  a  man  dig  in  10  hours  ?  How  many- 
bushels  in  10  days  of  10  hours  each  ? 

2.  In  measuring  depths  at  sea,  6  feet  are  a  fathom. 
How  many  feet  are  10  fathoms  ?     Are  100  fathoms  ? 

3.  When  the  street-car  fare  is  5  cents,  how  much  will 
10  rides  cost  ?     How  much  will  100  rides  cost  ? 

J/..  If  a  joiner  can  make  8  window-frames  in  a  week, 
how  many  can  he  make  in  10  weeks  ? 


INTEGERS.— MUL  TIPLICATION.  59 

5.  How  much  will  10  barrels  of  beef  cost,  at  $23  a 
barrel?     At  $27  a  barrel? 

6.  A  field  of  10  acres  of  wheat  yielded  36  bushels  to 
the  acre.     What  was  the  total  yield  ? 

7.  How  many  barrels  of  flour  will  a  teamster  draw  at 
86  loads,  if  he  draws  10  barrels  at  a  load  ? 

8.  32  quarts  are  a  bushel.  How  many  quarts  are  10 
bushels  ?     Are  100  bushels  ?     Are  1,000  bushels  ? 

9.  A  box  of  8  by  12  window-glass  contains  75  panes. 
How  many  panes  of  glass  are  there  in  10  boxes  ?  In  100 
boxes  ?     In  1,000  boxes  ? 

10.  There  are  98  pounds  of  flour  in  a  half-barrel.     How 
many  pounds  are  there  in  1,000  half -barrels  ? 

11.  "What  is  the  cost  of  10,000  sewing-machines,  at  $24 
apiece  ? 

How  is  a  number  multiplied 

12.  By  10?     Why? 

13.  By  10  X  10,  or  100  ?     Why  ? 
U.  By  10  X  100,  or  1,000  ?     Why  ? 

i5.  By  10,000?    By  100,000?    By  1,000,000?    Why? 


76.  How  much  is 

1.  3  times  5  barrels? 

2.  30  times  5  barrels? 

3.  300  times  5  barrels  ? 
Jf..  3  times  25  peaches? 

5.  30  times  25  peaches? 

6.  300  times  25  peaches? 


What  is  the  product  of 

7.  5  multiplied  by  3  ? 

8.  5  multiplied  by  30  ? 

9.  5  multiplied  by  300  ? 

10.  25  multiplied  by  3  ? 

11.  25  multiplied  by  30? 

12.  25  multiplied  by  300? 


13.  At  9  cents  a  box,  how  much  must  I  pay  for  4  boxes 
of  table  salt  ?     How  much  for  40  boxes  ? 

IJf.  At  the  rate  of  7  miles  an  hour,  how  many  miles  will 
a  boat  sail  in  10  hours  ?     In  60  hours  ? 


60 


SECOND    BOOK  IiV  ARITHMETIC. 


15,  If  16  poles  are  required  for  a  mile  of  telegraph  line, 
how  many  poles  will  be  required  for  10  miles?  How 
many  for  30  miles  ?     For  100  miles  ?     For  300  miles  ? 

16,  4  peeks  are  a  bushel.  How  many  pecks  are  10 
bushels?  50  bushels?  100 bushels?  500  bushels?  1,000 
bushels  ?     5,000  bushels  ? 

17,  How  much  will  9,000  cords  of  wood  cost,  at  $5  a 
cord? 


Written  Worlc. 


Process. 
291 
60 


17,1^60 


11'7.  Ex.  Multiply  291  by  60. 

Explanation. —  Since  60  is  10  times  6,  I 
multiply  291  by  6,  and  to  the  product  an- 
nex a  cipher. 

Pkoblems. 
In  each  of  the  next  eight  problems,  write  the  product 
without  writing  the  factors. 


1.  100  times  75  = 

2,  1,000  times  75  = 
S,  100  times  392  = 

.4.  1,000  times  1,839  = 

5. 
6. 
7. 
8, 

392x10,000  = 
2,893X10,000  = 
478x1,000,000  = 
1,000,000X58,054  = 

9                              10 

II 

667 

80 


682 
400 


1,908 

7,000 


\2 

6,947 

90,000 


17,398 

700,000 


[4 

90,086 

5,000,000 


15,  What  is  the  product  of  30  times  254? 

16,  Multiply  249  by  4,000.     By  4,000,000. 

17,  What  is  the  product  of  600  X -972?  Of  600,000x972? 

18,  Find  the  product  of  the  factors  90,000  and  2,165. 


INTEGERS.^MULTIPLICA  TION.  61 

^       19.  How  many  bushels  of  oats  will  800  horses  eat  in  a 
year,  allowing  183  bushels  for  each  horse  ? 

W.  A  tea  merchant  bought  700  chests  of  tea,  each  con- 
taining 42  pounds.  How  many  pounds  of  tea  did  he 
buy? 

21.  How  many  gallons  are  there  in  a  cargo  of  7,000 
barrels  of  kerosene,  of  42  gallons  each  ? 

22.  In  214  barrels  of  fish,  containing  200  pounds  each, 
are  how  many  pounds  ? 

23.  Sixty  minutes  are  one  hour.  How  many  minutes 
are  twenty-four  hours,  or  one  day  ? 

Find  the  cost 
2J,..  Of  300  hogsheads  of  molasses,  @  $65. 

25.  Of  125  barrels  of  pork,  @  $20. 

26.  Of  176  acres  of  land,  @  $50. 

27.  Of  70  bushels  of  wheat,  @  $1.25. 

28.  Of  500  bushels  of  corn,  @  $  .70. 

29.  Of  18  piano-fortes,  at  $270  each. 

30.  Of  150  chairs,  at  $  .65  each. 

31.  Of  600  sheep,  at  $4.25  a  head. 

Case  III.  The  multiplier  two  or  more  digits. 

Of'ol  Work. — 78.  A*  1.  At  8  cents  apiece,  how  much 
will  4  cocoa-nuts  cost?  How  much  will  20  cost?  How 
much  will  24  cost  ? 

2.  A  cooper  made  9  barrels  a  day  for  32  days.  How 
many  barrels  did  he  make  in  2  days  ?  How  many  in  30 
days  ?     How  many  in  the  32  days  ? 

3.  At  6  cents  for  the  use  of  1  dollar  for  a  year,  how 
much  must  I  pay  for  the  use  of  7  dollars?  How  much 
for  the  use  of  30  dollars  ?     For  the  use  of  37  dollars  ? 


62  SECOND    BOOK  IN   ARITHMETIC. 

How  much  is  the  cost 

^.  Of  84  pairs  of  boots,  at  $5  a  pair  ? 

6,  Of  69  yards  of  cassimere,  at  $2  a  yard  ? 

6,  Of  98  miles  of  railroad  fare,  at  $  .03  a  mile  ? 

7.  Of  54  reams  of  book  paper,  at  $7  a  ream  ? 

8.  What  is  the  product,  when  the  multiplicand  is  ones  ? 

9,  When  it  is  tens?  |  10.  Hundreds?  |  11,  Thousands? 

J5,  1,  5  times  25,  plus  10  times  25  are  how  many  times 
25? 

^.  Then,  how  is  25  multiplied  by  15  ? 

3,  How  is  any  number  multiplied  by  15  ? 

4'.  2  times  any  number,  plus  20  times  the  same  number 
are  how  many  times  the  number  ? 

6,  Then,  how  is  any  number  multiplied  by  22  ? 

6,  5  times  any  number,  plus  YO  times  the  same  number 
are  how  many  times  the  number  ? 

7.  Then,  how  is  any  number  multiplied  by  75  ? 
How  is  any  number  multiplied 


8,  By  36  ? 
a  By  84  ? 


10,  By  17? 

11.  By  91  ? 


12.  By  215? 

13.  Bv  975  ? 


i^.  By  4,215? 
15.  By  2,047  ? 


79.  Principle  III.  The  product  resulting  from  mul- 
tijplying  one  nurriber  hy  the  ones^  tens^  hundreds,  etc.,  of 
another,  is  the  sum  of  the  several  products. 

These  several  products  are  called  partial  products, 

1.  7  times  13,5284-40  times  13,528  +  2,000  times  13,528 
are  how  many  times  13,528  ? 

2.  7  +  40  +  2,000  times  13,528  are  how  many  times 
13,528  ? 


INTEGERS.— MUL  TIPLICA  TION.  63 

Written  Work.— 80.  Ex.  Multiply  13,528  by  2,047. 

Full  Process.  Common  Process. 

13,628  13,528 

9Jf696^  7time8l3,528=  91^,696 

6Jfll20=  40  times  13,528=         *  5  Jf.  1 1  2 


Partial 
products. 


27066000=  2,000  times  13.528=  27066 


A. 


27,691,816=  2,04T  times  13,528  =  27,691,816 

Explanation. — Writing  the  factors  with  the  ones  of  the  multi- 
plier under  the  ones  of  the  multiplicand,  I  multiply  first  by  7 
ones,  next  by  4  tens  or  40,  and  then — as  there  are  no  hundreds 
— by  2  thousands  or  2,000,  and  write  the  first  figure  of  each  par- 
tial product  under  the  figure  of  the  multiplier  used  to  obtain  it. 

I  then  add  the  partial  products,  and  obtain  27,691,816,  the  re- 
quired product. 

Problems. 

1  i  ill 

74  281  2,976  426  4,306 

23  54  81  315  284 


6  7  8  9  10 

13,008  50,028  4,765  29,872  8,009 

472  974  807  5,008  6,003 


a.  Multiplying  any  number  hy  0  produces  0. 
h.  Multiplying  0  hy  any  number  produces  0. 

11,  What  is  the  product  of  58  times  15Y? 

12,  73  X  593  .= how  many  ?     Multiply  17,248  by  9,005. 

13,  What  is  the  product  of  4,293  multiplied  by  2,726  ? 
i^.  308  times  203  times  69  yards  are  how  many  yards  ? 
16.  A  load  of  74  bushels  of  oats,  weighing  32  pounds  to 

the  bushel,  weighs  how  many  pounds  ? 


64  SECOND    BOOK  IN  ARITHMETIC, 

16,  In  a  barrel  containing  87  dozens  of  eggs,  are  how 
many  eggs  ? 

17,  A  lioe  factory  that  makes  1,396  hoes  per  week, 
makes  how  many  hoes  in  a  year,  or  52  weeks  ? 

18,  A  certain  daily  newspaper  office  uses  217  reams  of 
paper  per  day.  How  many  reams  does  it  use  in  the  308 
business  days  of  a  year  ? 

19,  If  one  mile  of  telegraph  wire  weighs  489  pounds, 
what  will  be  the  weight  of  the  wire  for  a  line  of  telegraph 
138  miles  long  ? 

J5  1  ^  i  1  1 

$31,075  $310.75  $50.44  $243.75  $1,274.09 
36                     36                 705                   215  8,047 

6,  A  wholesale  dealer  sold  80  watches,  at  $75  each. 
How  much  did  he  receive  for  them  ? 

7,  The  yearly  wages  of  307  men  in  a  coal  mine  are 
$736  each.  What  is  the  amount  of  their  wages  for  the 
year? 

8,  What  is  the  value  of  132  ounces  of  gold,  at  $16  an 
ounce  ? 

9,  One  year  the  repairs  on  a  canal  203  miles  long,  cost 
$382.75  per  mile.     What  was  the  total  expense  ? 

10,  What  is  the  value  of  the  735  horses  of  a  city  horse- 
car  line,  at  $149  each  ? 

11,  One  week  a  butcher  bought  356  lambs,  at  $3.56  a 
head.     How  much  did  they  cost  him  ? 

1^2,  At  $  .18  a  gallon,  what  is  the  cost  of  a  barrel  of 
kerosene  containing  42  gallons? 

13,  How  much  will  28  bushels  of  turnips  cost,  at  $  .31 
a  bushel  ? 


INTEGERS.^MUL  TIPLICA  TION.  65 

How  much  must  I  pay 

H.  For  369  pounds  of  wool,  @  $  .64? 

15.  For  32  yards  of  body-Brussels  carpeting,  @  $2.19  ? 

16.  For  23  rolls  of  paper-hangings,  @  $  .27? 

17.  A  milkman  sells  219  quarts  a  day,  at  %  .05  a  quart. 
How  much  do  his  sales  amount  to  in  a  year  ? 

Case  IY.  Each  factor  a  digit  or  digits,  with  a  ci- 
pher or  ciphers  on  the  right. 

Oral  Work. — 81o  Ao  Find  the  cost 

1.  Of  a  turkey  that  weighs  10  pounds,  at  $  .20  a  pound. 

2.  Of  50  pounds  of  sugar,  at  $  .10  a  pound. 

3.  Of  30  pounds  of  chickens,  at  $  .20  a  pound. 
Jf.  Of  20  pounds  of  salmon,  at  $  .30. 

6.  Of  30  barrels  of  apples,  at  150  cents,  or  $1.50. 

6.  Of  200  acres  of  land,  at  $10.     At  $20.     At  $50. 

7.  A  barrel  of  pork  weighs  200  pounds.  What  is  the 
weight  of  3  barrels  ?     Of  30  barrels  ?     Of  300  barrels  ? 

8.  There  are  160  acres  in  a  quarter-section  of  govern- 
ment land.  How  many  acres  are  in  4  quarter-sections? 
In  40  quarter-sections  ? 

9.  How  many  pounds  of  butter  are  there  in  40  fifty- 
pound  tubs  ?     In  400  fifty-pound  tubs  ? 

10.  One  year  a  grocer  sold  80  thirty-pound  tin  cans  of 
maple  sugar.     How  many  pounds  of  sugar  did  he  sell  ? 

1^,  How  many  are  What  is  the  product 

1.  3  times  20  windows  ?  ^.  Of  3  times  20,  or  20  times  3  ? 

S.  10  times  3  times  20  boards?  J,..  Of  30  times  20  ? 

5.  4  times  200  bricks  ?  ^  Of    4    times    200,   or    200 

7.   10  times  4  times  200  shin-  times  4  ? 

gles?  8.  Of  40  and  200? 

9.  100  times  3  times  50  rails?  10.  Of  50  and  300? 


66 


SECOND    BOOK  IN  ARITHMETIC. 


How  many  ciphers  are  on  the  right  of  both  factors,  and 
how  many  are  on  the  right  of  the  product 

11.  In  questions  1  and  2  ?     I     13.  In  questions  5  and  6  ? 

12.  In  questions  3  and  4?     |      IJf.  In  questions  7  and  8? 

15.  In  questions  9  and  10? 

82.  Principle  TV.  There  are  at  least  as  mam,y  ciphers 
on  the  right  of  a  product^  as  there  are  on  the  right  of  the 
factors  which  jproduce  it 

Written  Work.— Ex.  Multiply  7,200  by  160. 


Explanation.— Writing  the  factors  with  the 
right-hand  digit  of  the  multiplier  under  the 
right-hand  digit  of  the  multiplicand,  I  mul- 
tiply the  72  hundreds  of  the  multiplicand 
by  the  16  tens  of  the  multiplier,  and  obtain 
1,152.  To  this  I  annex  three  ciphers — one 
for  the  cipher  on  the  right  of  the  multi- 
plier, and  two  for  the  two  ciphers  on  the 
right  of  the  multiplicand. 

The  result,  1,152,000,  is  the  required  product. 


Pkocess. 

7,200 
160 
4,32 
72 
1,152,000 


2400 


1,760 
39 


Problems. 

3  4 

270  1,920 

960  650,000 


428,000 
720 


480 
7000 


9906432 
789600 


8 

4893700 
536000 


9 

596802 
307020 


10.  One  month  a  rolling-mill  made  5,800  bars  of  rail- 
road iron,  each  bar  weighing  402  pounds.  How  much  did 
the  whole  weigh  ? 

11.  At  the  rate  of  1,400  words  an  hour,  how  many 
words  can  be  sent  over  a  telegraph  line  in  30  hours  ? 


INTEGERS.— MUL  TTPLICA  TION.  67 

12,  How  much,  will  a  yearly  salary  of  $1,700  amount 
to  in  12  years  ? 

13.  How  mucli  will  it  cost  to  fence  800  miles  of  rail- 
road, at  $150  a  mile  ? 

H,.  If  a  rope-maker  can  spin  19,240  feet  of  rope  in  a 
week,  liow  many  feet  can  lie  spin  in  50  weeks  ? 

15.  A  barge  was  loaded  with  600  bales  of  hay,  weighing 
290  pounds  •€ach.     What  was  the  weight  of  the  load  ? 

16.  How  many  grape-vines  will  be  required  for  a  vine- 
yard of  40  acres,  allowing  1,280  vines  to  the  acre  ? 

17.  A  nursery-man  sold  2,800  young  apple-trees,  at  $  .17 
apiece.     How  much  did  he  receive  for  them  ? 

18.  How  much  will  50  bushels  of  peaches  cost,  at  $2.50 
a  bushel  ? 

19.  How  much  must  be  paid  for  transporting  450  tons 
of  freight  from  New  York  to  St.  Louis,  at  $27.50  a  ton? 

W.  In  a  barrel  of  beef  are  200  pounds.  How  many 
pounds  are  there  in  37  barrels  ? 

21.  A  planter  raised  94  acres  of  cotton,  which  yielded 
460  pounds  to  the  acre.     What  was  the  total  yield  ? 

M.  In  a  ream  of  paper  are  480  sheets.  How  many 
sheets  are  there  in  260  reams  ? 

23.  Eequired  the  cost  of  building  a  line  of  telegraph 
680  miles  long,  at  $1,250  a  mile. 

^4"  A  manufacturer  of  reapers  and  mowers  sold  in  one 
year  2,500  machines,  at  $130  each.  How  much  did  he 
receive  for  them  ? 

25.  The  factors  are  three  hundred  ninety-seven  thou- 
sand five  hundred,  and  nine  thousand  eight  hundred. 
What  is  the  product  ? 


68  SECOND   BOOK  IN  ARITHMETIC. 

83.  Rules  for  Multiplication  of  Integers. 

I.  The  multiplier  a  digit. 

1.  Write  the  inultijplier  under  the  ones  of  the  multi- 
pliccmd, 

2.  Beginning  at  the  right,  multiply  the  units  of  each 
order  in  the  multiplicand  by  the  multiplier  j  in  the  prod- 
uct write  the  ones  of  each  result,  a/ad  add  the  tens  to  the 
next  result, 

II.  The  multiplier  a  digit,  with  a  cipher  or  ciphers 

on  the  right. 

1.  Write  the  factors — the  digit  of  the  multiplier  under 
the  ones  of  the  multiplicand. 

2.  Multiply  the  multiplicand  hy  the  digit  of  the  multi- 
plier, and  to  the  product  annex  the  ciphers. 

When  the  digit  is  1 — i.e.,  when  the  multiplier  is  10,  100,  1,000,  etc., 
—perform  the  process  m^entaUy. 

III.  The  multiplier  two  or  more  digits. 

1.  Write  the  multiplier  under  the  multiplicand — ones 
under  ones,  tens  under  tens,  and  so  on, 

2.  Beginning  at  the  right,  multiply  the  multiplicand 
hy  the  ones,  tens,  hundreds,  etc,  of  the  multiplier,  place 
the  right-hand  figure  of  each  partial  product  under  the 
figure  of  the  multiplier  used  to  obtain  it,  and  add  the 
partial  products, 

IV.  Each  factor  a  digit  or  digits,  with  a  cipher  or 
ciphers  on  the  right. 

1.  Write  the  factors — the  right-hand  digit  of  the  multi- 
plier under  the  right-hand  digit  of  the  multiplicand, 

2.  Omitting  the  ciphers  on  the  right  of  the  factors,  mul- 
tiply as  in  Case  III ;  and  to  the  product  thus  obtained 
annex  the  ciphers. 


INTEGERS.-'MUL  TIPLICA  TION.  69 

Peoblems. 
A.  L  ^  1  ^  t 

1,026,348     1,327    2,076    175,941         9,654 
6      246      382         400,000     21,800 

6  2  ?.  i  '^ 

64                      68,000  49,500  1,850  53,276 

70,000  73 400  63,000  5,002 

Ex.  Multiply  654  by  9. 

Process,  Making  the  Computation. 

65  Jf,  9  4's  are  36;  write  6  (add  3). 

Q  9  5's  and  3  are  48;  write  8  (add  4). 

9  6's  and  4  are  58;  write  58. 

6^886  Result,  5,886. 

11,  An  excursion  train  of  9  cars  lias  84  passengers  in 
each  car.     How  many  passengers  are  on  the  train  ? 

12,  Every  mile  of  a  road  4  rods  wide  contains  8  acres. 
How  many  acres  are  there  in  168  miles  of  such  a  road  ? 

13,  If  290  pounds  of  cheese  are  made  from  the  milk  of 
one  cow,  how  many  pounds  can  be  made  from  the  milk 
of  470  cows  ? 

H.  If  a  miller  grinds  1 50  barrels  of  flour  in  a  day,  how 
many  barrels  will  he  grind  in  56  days  ? 

15,  How  many  tons  of  iron  will  be  required  for  359 
miles  of  railroad,  at  97  tons  to  the  mile  ? 

16,  How  much  will  282  street-cars  cost,  at  $825  each  ? 

17,  One  season  a  steamboat  on  the  Hudson  made  234 
trips,  and  the  average  number  of  persons  carried  per  trip 
was  108.  How  many  persons  did  the  boat  carry  during 
the  season? 

18,  How  many  feet  of  lumber  can  be  cut  from  1,000 
logs,  each  log  making  642  feet? 


70  SECOND   BOOK  IN  ARITHMETIC.  4 

Oral  Work. — B.  1.  Ten  kits  of  mackerel  of  twenty- 
five  pounds  each  are  how  many  pounds?  One  hundred 
kits  are  how  many  pounds  ? 

^.  A  furrier  sold  4  muffs  at  $30  apiece.  How  much 
did  he  receive  for  them  ? 

S.  How  much  must  I  pay  for  5  pounds  of  beefsteak,  at 
16  cents  a  pound  ? 

^.  How  many  pounds  of  butter  are  there  in  3  four-gal- 
lon jars  of  32  pounds  each  ? 

5.  How  many  miles  will  a  steamboat  run  in  24  hours, 
if  she  runs  8  miles  an  hour  ? 

6.  How  much  will  40  silk  hat^-  cost,  at  $6  apiece  ? 

7.  Six  spoons  are  a  set.  How  many  sjDOons  are  seventy- 
five  sets  ? 

8.  What  is  the  cost  of  20  paper-weights,  at  $  .60  each  ? 

Written  Work, — C.  Multiply 


t  7,945  and  70,087  by  7. 
^.  4,670  and  37,500  by  40. 


3.  4,386  by  9  ;  by  800  ;  by  3,764. 

4.  89,760  by  787 ;  by  20,960. 

7  8 


Multiplicands,    6,597        892,367  84,720  967,008 

Multipliers,  9  1,000  326,000         30,854 

Find  the  product  of  the  factors 


9.  3,156,    592,  and  8. 

10.  $5,120,    786,  and  25. 

11.  98,   $1.63,   39,  and  9. 


12.  47,    3,400,  and  $697.48. 

13.  90,876,    3,008,    90,  and  45. 
U.  $897.28,    500,    17,    36,  and  24. 

15.  If  245  pounds  of  charcoal  are  used  in  making  1 
ton  of  gunpowder,  how  many  pounds  are  used  in  making 
1,056  tons? 

16.  A  common  clock  strikes  156  times  every  day.  How 
many  times  does  it  strike  in  a  year  ? 

17.  A  hotel  keeper  bought  23  barrels  of  eggs,  each  bar- 
rel containing  83  dozen.     How  many  eggs  did  he  buy  ? 


INTEGERS.— M  UL  TIPLICA  TION.  71 

18.  A  mail  agent  made  225  round  trips  each  year  for 
13  years,  over  a  railroad  108  miles  long.  How  many 
miles  did  lie  travel  ? 

19.  24  sheets  of  paper  are  a  quire,  and  20  quires  are  a 
ream.     How  many  sheets  are  there  in  45  reams  ? 

W.  How  many  buttons  are  there  in  7  dozen  packages, 
each  package  containing  10  cards,  and  each  card  3  dozen 
buttons  1 

'21.  A  shoe  dealer  bought  45  cases  of  ladies'  French  kid 
boots,  each  case  containing  12  pairs,  at  $3  a  pair.  What 
was  the  amount  of  the  purchase  ? 

2%.  A  Pennsylvania  oil  well  flowed  237  days,  at  the  rate 
of  345  barrels  of  oil  per  day.    How  much  oil  did  it  yield  ? 

23.  A  manufacturer  sold  319  sets  of  chairs,  at  %  .65 
apiece.     How  much  did  he  receive  for  them  \ 

Note. — For  outlines  of  multiplication  for  review,  see  page  270. 

Eeview  Problems. 
Oral  Work. — 1.  A  man  built  a  fence  in  10  days,  build- 
ing 32  rods  each  day.    What  was  the  length  of  the  fence  ? 

2.  How  far  will  an  express  train  run  in  9  hours,  run- 
ning 34  miles  an  hour  ? 

3.  A  steamboat  plies  between  two  places  that  are  50 
miles  apart.     How  many  miles  does  she  run  in  12  trips  ? 

^.  If  a  house  rents  for  $60  a  month,  how  much  does 
the  rent  amount  to  in  10  months  ? 

5.  A  merchant  bought  5  pieces  of  cambric,  of  44  yards 
each,  at  10  cents  a  yard.     How  much  did  it  cost  him  ? 

6.  A  fruit  dealer  bought  20  crates  of  peaches  for  $48, 
and  sold  them  at  $3  a  crate.     How  much  was  his  gain  ? 

7.  Mabel  bought  10  yards  of  calico  at  13  cents  a  yard, 
and  8  yards  of  ribbon  at  20  cents  a  yard.  How  much  did 
her  purchases  amount  to  ? 


72  SECOND   BOOK  IN  ARITHMETIC. 

8.  8  fields,  of  80  acres  each,  contain  how  many  acres  ? 

9,  In  a  flouring  mill  are  8  pairs  of  millstones,  and  each 
pair  will  grind  100  bushels  of  wheat  per  day.  How  many 
bushels  of  grain  can  the  mill  grind  in  a  week  ? 

10.  A  hatter  bought  8  silk  hats  for  $40,  and  sold  them 
at  $6  apiece.     How  much  did  he  gain  ? 

Written  Work, — 1.  How  many  square  miles  are  there 
in  25  townships  of  36  square  miles  each  ? 

^.  In  a  factory  are  10  lines  of  shafting;  one  of  them 
weighs  882  pounds,  and  the  other  9  weigh  462  pounds 
each.     What  is  their  total  weight  ? 

S.  If  you  have  $127  when  you  are  16  years  old,  and 
you  save  $39  each  year  until  you  are  21,  how  much  mon- 
ey will  you  then  have  ? 

^.  A  book-keeper's  wages  are  $56  a  month,  and  his 
expenses  for  a  year,  or  12  months,  are  $608.  How  much 
does  he  save  in  a  year  ? 

5.  A  Government  surveyor  receives  $150  a  month, 
and  expends  $68.     How  much  does  he  save  in  a  year? 

6.  How  many  pickets  will  be  required  for  the  fence 
that  encloses  a  lot  198  feet  long,  and  143  feet  wide,  allow- 
ing 3  pickets  to  the  foot  ? 

7.  How  much  must  I  pay  for  25  thousand  feet  of  wal- 
nut lumber,  at  $41.81  per  thousand  ? 

8.  At  $6  per  week,  how  much  will  36  boarders  pay 
for  board  in  one  year,  or  52  weeks  ? 

9.  A  stock  train  consists  of  17  cars,  each  car  contain- 
ing 179  sheep.  How  much  do  all  the  sheep  weigh,  their 
average  weight  being  95  pounds  ? 

10.  A  merchant  bought  37  cases  of  prints,  each  case 
containing  12  pieces  of  42  yards  each.  How  many  yards 
of  print  did  he  buy  ? 


INTEGERS.— MULTIPLIC A  TION.  73 

11.  If  I  buy  40  horses  at  $120  each,  and  sell  all  of 
them  for  $5,000,  how  much  do  I  gain  ? 

12.  A  farmer  raised  2  fields  of  potatoes.  The  first 
field,  of  5  acres,  yielded  102  bushels  to  the  acre ;  and  the 
second  field,  of  8  acres,  yielded  119  bushels  to  the  acre. 
How  many  bushels  of  potatoes  did  he  raise  ? 

13.  A  manufacturer  pays  125  workmen  $39  a  month 
each.     How  much  do  their  wages  amount  to  in  a  year  ? 

H.  A  farmer  who  raised  1,221  bushels  of  oats,  kept  85 
bushels  for  seed,  and  enough  to  winter  9  horses,  allowing 
50  bushels  to  each  horse,  and  sold  the  balance.  How 
many  bushels  did  he  sell  ? 

15.  Of  5  hogsheads  of  sugar,  billed  at  1,125  pounds 
each,  the  first  was  72  pounds  short,  the  second  56  pounds, 
the  third  38  pounds,  the  fourth  112  pounds,  and  the  fifth 
47  pounds.     How  many  pounds  were  in  the  5  hogsheads  ? 

16.  A  merchant  bought  24  sets  of  crockery  of  45  pieces 
each,  29  sets  of  37  pieces  each,  and  36  sets  of  80  pieces 
each.     How  many  sets  did  he  buy  ?     How  many  pieces  ? 

17.  A  drover  bought  69  cattle,  at  $28.75  a  head.  He 
sold  27  of  them,  at  $37.75  a  head ;  and  the  remainder,  at 
$36.50  a  head.     Did  he  gain  or  lose,  and  how  much? 

18.  A  provision  dealer  bought  806  barrels  of  fish,  at 
$16.50  a  barrel ;  and  sold  the  lot  at  $1,847  more  than 
cost.     How  much  did  he  receive  for  it  ? 

19.  An  agent  sold  3  farms.  For  the  first  he  received 
$675,  for  the  second  twice  as  much  as  for  the  first,  and 
for  the  third  4  times  as  much  as  for  the  first  and  second. 
How  much  did  he  receive  for  the  second?  For  the 
third?     For  the  3  farms? 

Note. — Teachers  who  prefer  that  decimals  should  be  studied  in  connec- 
tion with  integers,  will  now  require  their  pupils  to  study  multipliqation 
of  decimals,  pages  126-129. 

D 


SECTION  V. 

DIVISION. 

Oral  Work. — 84.  How  many        How  many  times 

1.  Are  2-  2  ?  2,  Can  2  be  taken  from  2  ? 

3,  Are  4-2-2  ?  ^.  Can  2  be  taken  from  4  ? 

5.  Are  6  —  2  —  2  —  2?  6.  Can  2be  taken  from  6? 

7.  Are  8-2  —  2-2-2?  8.  Can  2  be  taken  from  8? 

9,  4  is  how  many  2's  ?  How  many  times  2  ? 
10.  6  is  how  many  2's?  How  many  times  2  ? 
ii.  8  is  how  many  2's  ?  How  many  times  2  ? 
i^.  How  many  times  is  2  contained  in  4  ?     In  6  ?     In  8  ? 

85.  To  divide  a  number  is  to  separate  it  into  equal 
parts. 

When  any  number  is  divided 

Into  two  equal  parts,  one  of  the  parts  is  one  half  of  the 
number ; 

Into  three  equal  parts,  one  of  the  parts  is  one  third  of 
the  number ; 

Into  four  equal  parts,  one  of  the  parts  is  one  fourth  of 
the  number ; 

Into  fi^e  equal  parts,  one  of  the  parts  is  one  fifth  of  the 
number ; 

Into  six  equal  parts,  one  of  the  parts  is  one  sixth  of  the 
number ;  and  so  on. 

Divide 

i.  2  into  2  equal  parts. 
2,  4  into  2  equal  parts. 

5.  6  into  2  equal  parts. 
Jf..   8  into  2  equal  parts. 

6.  3  into  3  equal  parts. 
^.12  into  3  equal  parts. 


7.  21  into  3  equal  parts. 

8.  8  into  4  equal  parts. 
^.24  into  4  equal  parts. 

10.  15  into  5  equal  parts. 

11.  40  into  5  equal  parts. 

12.  42  into  6  equal  parts. 


INTEGERS.— DIVISION. 


75 


86.  The  equal  parts  into  which  any  number  is  divided, 
are  commonly  0.2^^^  fractional  jparts^  ore  fractions. 
Halves,  thirds,  fourths,  and  fifths  are  written 

a.  To  find  one  half  of  a  number : — Divide  the  number  by  2, 
h.  To  find  one  third  of  a  number : — Divide  the  number  by  S, 
€.  To  find  one  fourth  of  a  number : — Divide  the  number  by  4* 
d.  To  find  one  fifth  of  a  number : — Divide  the  number  by  5. 

How  much  is 

5.  1  third  of  21?       9,  ^of  6? 

6.  1  fourth  of  8?  10.  |of  12? 

7.  1  fourth  of  24?  11.  |of  16? 

8.  1  fifth  of  15?  12.  I  of  40? 


1.  1  half  of  2  ? 

2.  1  half  of  4  ? 
S.  1  half  of  8  ? 
U.  1  third  of  3? 


IS.  i  of  54  ? 
U.  I  of  80  ? 

15.  1  of  72  ? 

16.  I  of  42  ? 


87.  A.  1.  When  2  quarts  of  milk  cost  12  cents,  how 
much  does  1  quart  cost  % 

1  quart  costs  1  half  as  much  as  2  quarts,  and  1  half  of  12 
cents  is  6  cents. 

2.  2  pounds  of  crackers  cost  16  cents.  How  much  does 
1  pound  cost  % 

3.  I  paid  $12  for  6  barrels  of  potatoes.  What  was  the 
price  per  barrel  ? 

If,.  If  a  steam-engine  in  a  factory  burns  27  tons  of  coal 
in  3  days,  how  many  tons  does  it  burn  in  1  day  ? 

5.  In  a  garden  are  20  trees  in  4  equal  rows.  How  many- 
trees  are  in  each  row  ? 

6.  If  I  feed  my  chickens  1  bushel,  or  32  quarts,  of  corn 
in  4  days,  how  many  quarts  will  I  feed  them  in  a  day  ? 

7.  If  I  divide  25  almonds  equally  among  5  children^ 
how  many  almonds  shall  I  give  to  each  child  ? 


76  SECOND   BOOK  IN  ARITHMETIC. 

IB.  1.  At  2  cents  apiece,  how  many  peaches  can  I  buy 
for  8  cents  ? 

First  Solution. — At  2  cents  apiece,  I  can  buy  as  many  peaches 
for  8  cents  as  the  times  2  cents  can  be  taken  from  8  cents.  2 
cents  from  8  cents  leave  6  cents,  2  cents  from  6  cents  leave  4 
cents,  2  cents  from  4  cents  leave  2  cents,  and  2  cents  from  2 
cents  leave  0.  Since  2  cents  can  be  taken  from  8  cents  4  times, 
I  can  buy  4  peaches.      Or, 

Second  Solution. — At  1  cent  apiece,  for  8  cents  I  can  buy  8 
peaches;  and  at  2  cents  apiece,  for  8  cents  I  can  buy  1  half 
of  8  peaches,  Tvhich  is  4  peaches. 

^.  How  many  boys'  suits  can  be  made  from  18  yards  of 
cloth,  allowing  o  yards  for  a  suit  ? 

S,  How  many  days  will  20  pounds  of  flour  last  a  family 
that  uses  4  pounds  a  day  ? 

4'.  How  much  shall  I  receive  for  24  pairs  of  hosv%  at  the 
rate  of  4  pairs  for  a  dollar  ? 

5.  If  I  drive  my  horse  5  miles  an  hour,  how  many  hours 
will  it  take  me  to  drive  him  15  miles  ? 

88.  Division  is  the  process  of  separating  a  number 
into  equal  parts. 

a.  The  number  to  be  separated  into  equal  parts  is  the  dividend, 

b.  The  number  that  expresses  how  many  equal  parts  the  dividend 

is  to  be  separated  into,  is  the  divisor » 

c.  The  number  obtained  by  dividing  is  the  quotient. 

When  the  numbers  to  be  used  for  dividend  and  divisor  are 
like  numbers,  the  process  of  division  is  a  short  method 
of  subtraction. 

EXEECISES. 

A,  1.  The  dividend  is  54  and  the  divisor  is  6.  What 
is  the  quotient? 

^.  What  is  the  quotient  of  63  divided  by  7? 

S.  If  32  yards  of  carpeting  be  divided  into  breadths  of  6 
yards  each,  how  many  yards  will  there  be  of  the  remnant  ? 


INTEGERS.-^DIVISION. 


77 


4'.  56  loaves  of  bread  are  liow  many  times  8  loaves  ? 

5.  How  many  times  can  you  take  5  cents  from  38 
cents,  and  how  many  cents  will  remain  ? 

6.  Divide  $40  into  8  equal  parts. 

7.  Which  number  is  the  dividend,  which  the  divisor, 
and  which  the  quotient,  in  exercise  2  ?  In  exercise  4  ?  In 
exercise  6  ? 

8.  In  which  of  the  exercises  1-6  is  there  no  remainder  ? 

9.  What  kind  of  a  number  is  the  quotient,  in  exercise 
2  ?     In  exercise  4  ?     In  exercise  6  ? 


B.  Divide  by  2 

every  second  number 

1,  From  2  to  20. 
^.  From  20  to  2. 

Thus,  2  in  2  once,  2  iii  4  2 
times,  and  so  ou. 

2  in  20  10  times,  2  in  18  9 
times,  aud  so  on. 

Divide  by  3 
every  third  number 

5.  From  3  to  30. 

6.  From  30  to  3. 


Divide  into  2  equal  parts 

every  second  number 

S.  From  2  to  20. 

4.  From  20  to  2. 

Tbns,  1  half  of  2  is  1, 1  half 
of  4  is  2,  and  so  on. 

1  half  of  20  is  10,  1  half  of 
18  is  9,  and  so  on. 

Divide  into  3  equal  parts 
every  third  number 

7.  From  3  to  30. 

8.  From  30  to  3. 


89.  The  sign  of  division  is  a  short  horizontal  line 
with  a  dot  above,  and  a  dot  below  it,  -7-.  It  is  read  di- 
vided hy. 

Division  is  also  expressed  by  writing  the  dividend  above, 
and  the  divisor  below,  a  horizontal  line. 

72 
'72-^6  and  —  are  each  read  *"72  divided  by  6." 

Read 


1.  99^9  =  11 

2,  567^81=7 


3.  $513^3  =  $171 

4.  81-^9=z:27^3 


^•t- 


78 


SECOND    BOOK  IN  ARITHMETIC. 


Written  Work. — A»  Use  the  proper  sign,  and  write 

1.  The  quotient  of  18  divided  by  3  is  6. 

^.  1  fourth  of  60  quarts  is  15  quarts. 

3.  42  divided  by  6  equals  7. 

^.  $11.25  divided  by  5  equals  $2.25. 

5.  78  bushels  divided  by  6  equals  13  bushels. 

6,  Dividend,  120 ;  divisor,  12 ;  quotient,  10. 

jB.  Copy  and  complete 
1.  86-^-9=       5.  45  melons-T-5  = 
^.56  —  8=       6.  36  pears-r-4      = 
S,  42^6=        7.  36  apples-r-4    = 
4.  10-^1=       8.  72  cherries 4- 8  = 

Oral  Work. — 90,  A.  Divide 

by  4  every  fourth  number 

1.  From  4  to  40. 


melons 
pears 
apples 
cherries 


2.  From  40  to  4. 
Divide  by  5 

every  fifth  number 

5.  Fi-om  5  to  50. 

6.  From  50  to  5. 

Divide  by  6 

every  sixth  number 
9.  From  6  to  60. 
10.  From  60  to  6. 


9.  $80-^4=$ 

10.  $.40-^5=$ 

11.  $1.50^3  =  $ 

12.  $12.40-T-2=$ 

Divide  into  4  equal  parts 

every  fourth  number 

S.  From  4  to  40. 

Jf..  From  40  to  4. 

;  divide  into  5  equal  parts 
every  fifth  number 

7.  From  5  to  50. 

8.  From  50  to  5. 

Divide  into  6  equal  parts 
every  sixth  number 

11.  From  6  to  60. 

12.  From  60  to  6. 


S.  I  divided,  equally,  12  plums  between  2  girls,  12  pears 
among  3  girls,  12  peaches  among  4  boys,  and  12  apples 
among  6  boys. 

1.  Each  girl  had  what  part  of  the  12  plums  ?  Of  the 
12  pears  ? 

2.  Each  girl  had  how  many  plums  ?     How  many  pears  ? 

3.  Each  boy  had  what  part  of  the  12  peaches  ?  Of  the 
12  apples? 


INTEGERS.—DIVISION. 


79 


Jf,.  Each  boy  had  how  many  peaches  ?  How  many  apples  ? 

5.  If  9  curtains  cost  $36,  w^hat  part  of  $36  does  1  cur- 
tain cost  ?     How  many  dollars  does  1  curtain  cost  ? 

6.  One  seventh  of  a  farm  of  70  acres  is  woodland. 
How  many  acres  are  woodland  ? 

7.  Five  boys  gathered  forty  quarts  of  chestnuts,  which 
they  shared  equally.     How  many  quarts  had  each  boy  ? 

8.  How  many  hours  must  a  man  work  each  day,  to  do 
80  hours'  work  in  8  days  ?     In  10  days  ? 

9.  $  .45  for  9  quarts  of  currants  is  how  much  a  quart  ? 

10,  $56  for  8  weeks'  board  is  how  much  a  week  ?    How 
much  a  day? 

11,  100  bushels  of  wheat  from  10  acres  are  how  many 
bushels  to  the  acre  ? 

91.  A.  Copy,  complete,  learn,  and  recite  the 
Table  of  Primary  Combinations  in  Division. 


24-2  = 

12-4-3  = 

24-f-8  = 

40-^8  = 

3^3  = 

12-^4  = 

24-^4  = 

42-^6  = 

4^2  = 

14-4-2  = 

24-^6  = 

42-4-7  = 

4-^4  = 

14-4-7  = 

25-^5  = 

45-4-5  = 

5^5  = 

15-4-3  = 

27^3  = 

45-^9  = 

6^2  = 

15-^5  = 

27-4-9  = 

48-4-6  = 

6  —  3  = 

16-4-2  = 

28-4-4  = 

48-T-8  = 

6-^6  = 

16-4-8  = 

28-~7  = 

49-4-7  = 

7-^7  = 

16-f-4  = 

30^5  = 

54-^6  = 

8-^2  = 

18^2  = 

30-^-6  = 

54-4-9  = 

8-h4  = 

l8-^9  = 

32-4-4  = 

56^7  = 

8-8  = 

18-4-3  = 

32-4-8  = 

56-^8  = 

9-T-3  = 

18-r-6  = 

35^5  = 

63-4-7  = 

9-^9  = 

20-4-4  = 

35-^-7  = 

63^9  = 

10-^2  = 

20^5  = 

36^4  = 

64-4-8  = 

10-4-5  = 

21^3  = 

36^9  = 

72^8  = 

12-2  = 

21-^-7  = 

36-^6  = 

72-4-9  = 

12-4-6  = 

24^3  = 

40^5  = 

81-^9  = 

80 


SECOND    BOOK  IN  ARITHMETIC. 


B.  Copy,  complete,  leam,  and  recite  the 
Table  of  Equal  Parts. 


^oi    2  is 

I  of  12  is 

I  of  24  is 

1  of  40  is 

•J^  of    3  is 

1  of  12  is 

\  of  24  is 

1  of  42  is 

1^  of    4  is 

1  of  14  is 

1  of  24  is 

\  of  42  is 

^  of    4  is 

\  of  14  is 

\  of  25  is 

j  of  45  is 

1  of    5  is 

j  of  15  is 

I  of  27  is 

I  of  45  is 

1^  of    6  is 

\  of  15  is 

I  of  27  is 

I  of  48  is 

i  of    6  is 

|of  16  is 

1  of  28  is 

.  1  of  48  is 

^  of    6  is 

1  of  16  is 

4-  of  28  is 

\  of  49  is 

^  of    7  is 

^  of  16  is 

j  of  30  is 

j  of  54  is 

i-  of    8  is 

1  of  18  is 

I  of  30  is 

^  of  54  is 

1  of    8  is 

|of  18  is 

\  of  32  is 

\  of  56  is 

■J  of    8  is 

1  of  18  is 

i  of  32  is 

i  of  56  is 

1  of    9  is 

\  of  18  is 

\  of  35  is 

4^  of  63  is 

1  of    9  is 

1  of  20  is 

1  of  35  is 

i  of  63  is 

i  of  10  is 

\  of  20  is 

J  of  36  is 

-J  of  64  is 

\  of  10  is 

1  of  21  is 

\  of  36  is 

I  of  72  is 

I  of  12  is 

1  of  21  is 

\  of  36  is 

i  of  72  is 

1  of  12  is 

-J-  of  24  is 

^  of  40  is 

I  of  81  is 

OS.  Divide  by  7 
every  seventh  number 

1.  From  7  to  70. 

2.  From  70  to  7. 

Divide  by  8 

every  eighth  number 

5.  From  8  to  80. 

6,  From  80  to  8. 

Divide  by  9 

every  ninth  number 

9.  From  9  to  90. 

10.  From  90  to  9. 


Divide  into  7  equal  parts 

every  seventh  number 

3.  From  7  to  70. 

^  From  70  to  7. 

Divide  into  8  equal  parts 
every  eighth  number 

7.  From  8  to  80. 

8.  From  80  to  8. 

Divide  into  9  equal  parts 
every  ninth  number 

11.  From  9  to  90. 

12.  From  90  to  9. 


INTEGERS.— Dl  VISION.  81 

93,  Exercises  in  Division  at  Sight. 

2)  6    12    18     4    10    16     2     8    14    20 

3)  9  15  21  6  18  12  24  3  30  27 

4)16  8  40  12  24  32  20  36  4  28 

5)45  15  35  40  10  20  50  30  25  5 

6)24  30  42  36  48  12  54  18  6  60 

7^56  28  49  21  VO  35  42  7  63  14 

8)80  48  16  bQ  32  40  24  72  64  8 

9)72  36  54  9  63  81  45  27  18  90 

Note. — The  preceding  exercises  are  to  be  written  upon  the  board,  and 
used  in  class  drill  daily,  until  every  pupil  can  give,  at  sight,  the  result  in 
any  one  of  the  exercises. 

Oral  Work, — 94.  A.  1.  How  many  five-dollar  bills 
will  pay  for  a  cow  tliat  costs  forty  dollars  ? 

'2.  At  the  rate  of  4  quarts  an  hour,  in  how  many  hours 
will  a  boy  pick  32  quarts  of  strawberries  %    40  quarts  ? 

S.  Last  winter  Eliza  attended  school  50  days — 5  days 
each  week.     How  many  weeks  did  she  attend  school  ? 

^.  A  carpenter  built  a  house  in  54  days.  How  many 
weeks  was  he  in  building  it  ? 

5.  When  rice  is  7  cents  a  pound,  how  many  pounds  can 
be  bought  for  35  cents  ? 

6.  At  the  rate  of  9  plums  for  a  cent,  how  much  must  I 
pay  for  36  plums  ?     For  54  plums  ? 

-B,  1.  $80  for  10  stoves  is  how  much  for  1  stove  ? 
2.  70  cents  for  7  slates  is  how  much  for  1  slate  ? 
D2 


82 


SECOND    BOOK  IN  ARITHMETIC, 


S,  A  blacksmith  used  32  nails  in  setting  8  horseshoes. 
How  many  nails  did  he  use  for  each  shoe  ? 

^.  In  an  orchard  are  90  peach-trees  in  9  equal  rows. 
How  many  trees  are  there  in  a  row  ? 

5.  A  woman  bought  a  sewing-machine  for  $36,  and  paid 
for  it  in  6  equal  payments.    How  much  was  each  payment  ? 

6,  One  man  can  do  a  certain  piece  of  work  in  45  days. 
In  how  many  days  can  5  men  do  the  same  work  ? 

95.  Factor  or  Division  Table. 

32  is  8  times  4,  or  4  times  8. 
35  is  7  times  5,  or  6  times  7. 


4 
6 

8 

9 

10 

12 

14 
15 
16 


20 
21 
24 

25 

27 
28 

30 


s  2  times  2. 

s  3  times  2,  or  2  times  3. 

s  4  times  2,  or  2  times  4. 

s  3  times  3. 

s  5  times  2,  or  2  times  5. 

\  6  times  2,  or  2  times  6 ; 

\  4  times  3,  or  3  times  4. 
s  7  times  2,  or  2  times  7. 
s  5  times  3,  or  3  times  5. 
s  8  times  2,  or  2  times  8,  or 
4  times  4. 

j  9  times  2,  or  2  times  9 ; 

(  6  times  3,  or  3  times  6. 

j  10  times  2,  or  2  times  10; 

(    5  times  4,  or  4  times  5. 
s  7  times  3,  or  3  times  7. 

j  8  times  3,  or  3  times  8 ; 

(  6  times  4,  or  4  times  6. 
s  5  times  5. 

s  9  times  3,  or  3  times  9. 
s  7  times  4,  or  4  times  7. 

\  10  times  3,  or  3  times  10 ; 

'    6  times  5,  or  5  times  6. 


40  is  ■ 


36  is  9  times  4,  or  4  times  9,  or 
6  times  6. 

10  times  4,  or  4  times  10; 
8  times  5,  or  5  times  8. 
42  is  6  times  7,  or  7  times  6. 
45  is  9  times  5,  or  5  times  9. 

48  is  8  times  6,  or  6  times  8. 

49  is  7  times  7. 

50  is  10  times  5,  or  5  times  10. 
54  is  9  times  6,  or  6  times  9. 
56  is  8  times  7,  or  7  times  8. 
60  is  10  times  6,  or  6  times  10. 

63  is  9  times  7,  or  7  times  9. 

64  is  8  times  8. 

70  is  10  times  7,  or  7  times  10. 
72  is  9  times  8,  or  8  times  9. 

80  is  10  times  8,  or  8  times  10, 

81  is  9  times  9. 

90  is  10  times  9,  or  9  times  10. 
100  is  10  times  10. 


Note. — The  numbers  in  this  table  are  to  be  written  upon  the  board,  and 
used  In  class  drill  daily,  until  every  pupil  can  give,  at  sight,  the  factors  of 
any  number  in  the  table. 


INTEGERS.— DIVISION,  83 


1,  Divide  72  feet  by  9  feet. 


1  foot  is  contained  in  72  feet  72  times ;  and  9  feet  are  con- 
tained in  72  feet  1  ninth  of  72  times,  -which  is  8  times. 


Find 

7.  1  seventh  of  35  cents. 

8.  1  fifth  of  30  quarts. 

9.  1  fourth  of  40  pounds. 

10.  1  eighth  of  $40. 

11,  1  tenth  of  80. 


Divide 

^.32  nails  by  4  nails. 
S.  90  trees  by  10  trees. 
Jf..  45  days  by  9  days. 

5.  $36  by  $6. 

6,  70  by"  7. 

12.  In  which  of  the  preceding  exercises  are  the  quo- 
tients abstract  numbers? 

13.  In  which  are  the  quotients  concrete  numbers  ? 

Pkinciple  I.  The  dividend  and  quotient  are  always 
like  members. 

The  divisor  must  always  he  regarded  as  an  abstract  number, 

"What  is  the  quotient  of 


U.  $36  divided  by  $4  ? 

15.  $120  divided  by  $30? 

16.  $.75  divided  by  $.15? 

17.  150  cents  divided  by  25  cents? 


18.  $5  divided  by  $  .20  ? 

19.  $25  divided  by  $.50? 

20.  $2.40  divided  by  $  .08  ? 

21.  $12.50  divided  by  $1.25? 


If  the  given  divisor  is  cents,  and  the  dividend  dollars,  or  dol- 
lars and  cents,  reduce  the  dividend  to  cents  before  dividing. 

Case  I.  Tlie  divisor  ending  with  a  digit. 

96,  A.  1.  A  boy  earned  80  cents  in  4  days.      How 
much  did  he  earn  in  1  day  ? 

In  one  day  he  earned  1  fourth  as  much  as  in  four  days ;  and 
1  fourth  of  80  cents  is  20  cents. 

2.  $90  for  3  cows  is  how  much  for  1  cow  ? 

3.  120  hours  of  work  in  10  days  is  how  many  hours 
per  day? 

^.  128  cents  for  4  ducks  is  how  much  for  1  duck  ? 


84  SECOND    BOOK  IN  ARITHMETIC. 

6.  A  fanner  received  $140  for  wood,  at  $7  a  cord.  How 
many  cords  did  he  sell  ? 

At  $1  a  cord,  for  $140  he  would  sell  140  cords ;  aiid  at  $7  a 
cord,  for  $140  he  sold  1  seventh  of  140  cords,  which  is  20  cords. 

6.  If  a  factory  girl  can  weave  180  yards  of  carpeting  in 
6  days,  liow  many  yards  can  she  weave  in  a  day  ? 

7.  A  teacher  paid  $3.60  for  writing-books,  at  $.09 
apiece.     How  many  books  did  she  buy  ? 

8.  How  many  suits  of  clothes  can  be  made  from  84 
yards  of  cloth,  allowing  4  yards  to  a  suit  ? 

9.  A  farm  of  405  acres  is  to  be  divided  equally  among 
5  heirs.     How  many  acres  will  each  heir  receive  ? 

-B.  1.  If  a  boy  can  gather  96  bushels  of  apples  in  6 
days,  how  many  bushels  can  he  gather  in  1  day  ? 

He  can  gatlier  1  sixth  as  many  bushels  in  one  day  as  in  six 
days ;  I.  e.,  1  sixth  of  96  bushels. 

96  bushels  are  60  bushels  plus  36  bushels;  1  sixth  of  60 
bushels  is  10  bushels ;  1  sixth  of  36  bushels  is  6  bushels ;  and 
10  bushels  plus  6  bushels  are  16  bushels. 

^.  A  merchant  sold  144  spoons  in  sets  of  6  spoons  each. 
How  many  sets  of  spoons  did  he  sell  ? 

3.  How  many  cars  will  be  required  to  transport  280 
tons  of  coal,  at  8  tons  to  the  car  ? 

4^.  A  farmer  put  up  272  gallons  of  cider  in  8  casks. 
How  many  gallons  did  he  put  into  each  cask  ? 

5.  A  gardener  received  $4.95  for  radishes,  at  $.05  a 
bunch.     How  many  bunches  did  he  sell  ? 

6.  $276  for  4  acres  is  how  much  for  1  acre  ? 

7.  A  printer  uses  8  sheets  of  paper  in  making  a  book 
of  384  pages.  How  many  pages  of  the  book  does  1  sheet 
make? 

8.  If  9  pounds  of  crushed  sugar  cost  $1.35,  how  much 
does  1  pound  cost  ? 


INTEGERS.— DIVISION.  85 

9,  184  yards  for  8  silk  dresses  is  how  many  yards  for 
1  dress  ? 

10,  1,080  bushels  of  oats  from  20  acres  is  how  many 
bushels  from  an  acre  ? 

a  1,  1  fourth  of  80,  plus  1  fourth  of  8  is  1  fourth  of 
what  number  ? 

^.  60  divided  by  3,  plus  9  divided  by  3  equals  what 
number  divided  by  3  ? 

S,  \  of  140,  plus  I  of  28  equals  \  of  what  number  ? 

^.  80-^8,  plus  48-^8  equals  what  number  divided  by  8  ? 
Into  what  two  parts  will  you  separate  the  dividend, 


5.  In  finding!- of  180? 

6.  \  of  $2.22  ? 

7.  \oi  168  pails? 

8.  \  of  384  brooms? 


9.  In  dividing  130  by  2? 

10.  $275  by  $5? 

11.  864  yards  by  9  yards  ? 

12.  546  pens  by  7  pens  ? 


J>.  What  is  each  partial  dividend,  and  what  each  re- 
mainder, in  dividing 


1.  215  soldiers  into  8  squads? 

2.  $4.32  into  5  equal  parts? 


S.  531  acres  by  6  acres? 

^.  372  pounds  by  9  pounds? 


In  problem  1,  of  what  order  of  units  is 


5.  The  first  partial  dividend  ? 

6.  The  first  quotient  ? 

7.  The  first  remainder? 


8.  The  second  partial  dividend  ? 

9.  The  second  quotient  ? 
10.  The  second  remainder? 

Note. — Ask  similar  questions  on  each  of  the  problems  2,  3,  4. 
When  a  quotient  contains  two  or  more  orders  of  units,  the  part  of 
the  dividend  used  to  obtain  the  units  of  any  order  is  a  par- 
tial dividend. 

Written  Work.— 97.  Ex.  Divide  936  by  3. 

The  divisor  is  commonly  written  at  ^.  .  q\  o  o  /^  ^.  ■■,  -, 
XI  1  rx  p.-u  J-  -J  J  -XT-  1-  X  Divisor,  J  )  i' J  O  Dividend. 
the  left  01  the  dividend,  with  a  short  / 

vertical  curve  between  them.  312  Quotient. 


8     3  cents) 639  cents     5)2,055     4)1,648 


86  SECOND    BOOK  IN  ARITHMETIC. 

6  7  8  9 

8)1,648    2)840  papers    6)1,260  bricks    Y  tons) 3,542  tons 

98.  Ex.  Divide  36,824  by  8.  Fikst  Process. 

Explanation.  —  1  eighth  of  36  thou-  8\3 6^82 Jf,(Jt,^6 0 3 
sand,  the  first  partial  dividend,  is  4  q  (p 

thousand  and  a  remainder  ;  and  I 

write  4  for  the  thousands  of  the  quo-  Jf,  8 

tient.  4  thousand  times  8  are  32  thou-  /  g 

sand,  the  part  of  36  thousand  divided ; 


82  thousand  from  36  thousand  leaves  ^  ^ 

4  thousand,  the  part  of  36  thousand  2  L 

undivided;   and  4  thousand  (  =  40  

hundred ),  plus  the  8  hundred  of  the 

given  dividend  is  48  hundred,  the  g^^^^  p^^^^^ 

second  partial  dividend. 

1  eighth  of  48  hundred  is  6  hundred;  ^  ^36^8^  j, 

and  I  write  6  for  the  hundreds  of  the  L  60  3 

quotient.     6  hundred  times  8  are  48 
hundred,  the  partial  dividend  used; 

and  as  0  hundreds  remain  undivided,  the  2  tens  of  the  given 
dividend  is  the  third  partial  dividend. 

1  eighth  of  2  tens  is  0  tens ;  and  I  write  0  for  the  tens  of  the  quo- 
tient. The  2  tens  (=20)  undivided,  plus  the  4  ones  of  the  given 
dividend,  is  24  ones,  the  fourth  partial  dividend. 

1  eighth  of  24  is  3  ;  and  I  write  3  for  the  ones  of  the  quotient. 
3  times  8  are  24,  the  partial  dividend  used;  and  there  is  no 
remainder. 

The  result,  4,603,  is  the  required  quotient. 

In  the  first  process,  the  result  of  each  step  is  written.     This 
process  is  called  long  division. 

In  the  second  process,  only  the  final  result  or  quotient  is  writ- 
ten.    This  process  is  called  short  division. 

Long  division  may  be  used  with  any  divisor,  large  or  small ;  short 
division  should  be  used  whenever  the  divisor  is  expressed  by 
one  figure. 


INTEGERS.— DIVISION.  87 

99,  The  steps,  in  finding  the  units  of  each  order  in  a 
quotient,  are 

Ist.  Dividing f  to  find  the  number  of  units. 
2d.  Multiplying,  to  find  the  part  of  the  dividend  divided. 
M.  Subtracting,  to  find  the  part  of  the  dividend  undivided. 
4:th.  Annexing  to  the  remainder  the  units  of  the  next  lower 
order  of  the  dividend,  to  form  the  next  partial  dividend. 

Note.  —  In  long  division  both  divisor  and  i_  2 

quotient  may  be   written  at  the   right  of  the  fiBacisc  273 

dividend;  or  the  quotient  may  be  written  over  — —        25)6825 

the  dividend,  and  the  divisor  at  the  left,  as  here  — -    '  50 

shown  (1,2).    In  1,  the  factors  of  each  partial  ^  Ysk 

dividend  used  are  brought  near  together,  and  — ^  175 

no  space  is  occupied  at  the  left  of  the  dividend.  75 

In  2,  the  place  in  which  each  figure  of  the  quo-  —  7_5 
tient  stands  determines  its  local  value. 

Problems. 

Solve  the  first  14  of  the  following  problems  by  long 
division,  and  then  by  short  division. 

1.  If  4  tons  of  coal  are  used  each  day  in  an  iron  found- 
ery,  how  many  days  will  3,284  tons  last  ? 

^.  How  many  days  will  it  take  a  girl  to  braid  1,203 
straw  hats,  if  she  braids  3  hats  each  day  ? 

3.  A  city  fire  department  bought  3  fire-engines  for 
$14,064.     What  was  the  cost  of  each  ? 

Jf,.  Six  men  owning  a  coal  bed,  sold  it  to  nine  others  for 
$39,150.  How  much  did  each  seller  receive,  and  how 
much  did  each  buyer  pay  ? 

5.  If  you  buy  a  store  for  $3,624,  and  pay  for  it  in  3 
equal  yearly  payments,  how  much  will  you  pay  each  year  ? 

6.  A  plantation  of  1,845  acres  was  divided  equally 
among  9  heirs.     How  many  acres  did  each  heir  receive  ? 

7.  A  business  block  of  T  stores  cost  $8,652.  How  much 
did  each  store  cost  ? 


88  SECOND    BOOK  IN  ARITHMETIC. 

8.  An  iron  founder  received  $12.55  for  castings,  at 
$  .05  a  pound.     How  many  pounds  did  he  sell  ? 

9.  An  estate  of  $32,184  was  shared  equally  among  4 
heirs.     What  was  the  share  of  one  heir  ? 

10.  $4,627  is  the  cost  of  how  many  tons  of  coal,  at  $7  a 
ton? 

At  $1  a  ton  $4,627  is  the  cost  of  Process. 

4,627  tons;  and  at  $7  a  ton       $^,^^7=  Jt.,627  tons. 
$4,627  is  the  cost  of  1  seventh  z»^/v 

of   4,627  tons,  which   is    661        7)  ^^6^7_tons. 
tons.  661  tons. 

11.  A  boatman  carried  5,688  barrels  of  Onondaga  salt 
to  IS'ew  York,  in  6  loads.  How  many  barrels  did  he  take 
in  each  load  ? 

12.  In  how  many  days  will  a  cooper  make  2,712  barrels, 
if  he  makes  8  barrels  each  day  % 

IS.  If  a  teamster  draws  5  loads  of  freight  daily,  in  how 
many  days  will  he  draw  1,215  loads  ? 

i^.  How  many  miles  must  I  drive  my  horse  each  day, 
to  drive  222  miles  in  6  days  ? 

15.  My  front  fence  is  10  rods  long,  and  it  cost  me  $130. 
How  much  did  it  cost  per  rod  ? 

16.  A  stable  keeper  pays  $576  for  the  hay  to  winter  32 
horses.     What  does  the  hay  for  one  horse  cost  ? 

17.  10,304  pounds  of  corn  are  how  many  bushels  of  56 
pounds  each  % 

18.  How  many  cars,  each  carrying  48  passengers,  will 
be  required  to  carry  384  passengers  ? 

19.  If  the  cost  of  constructing  42  miles  of  railroad  is 
$2,032,632,  what  is  the  cost  of  constructing  1  mile  ? 

W.  In  how  many  days  can  a  paper-mill  make  10,080 
reams  of  note  paper,  if  it  makes  96  reams  a  day  ? 


INTEGERS,— DIVISION-.  89 

^1.  A  manufacturer,  having  1,096  ounces  of  silver,  made 
from  it  as  many  coffee  urns  as  possible,  each  weighing  45 
ounces,  and  a  salver  of  the  silver  that  he  had  left.  How 
much  did  the  salver  weigh  ? 

2^.  A  miller  packed  13,475  pounds  of  flour  in  sacks, 
putting  49  pounds  into  each.  How  many  sacks  did  he 
fill? 

23.  How  much  must  a  man  earn  in  each  of  the  313 
working  days  of  a  year,  to  earn  $1,252  ? 

2^.  A  butcher  paid  $11,616  for  352  beeves.  What  was 
the  cost  per  head  ? 

25.  Into  how  many  farms,  of  156  acres  each,  can  798 
acres  of  land  be  divided  ? 

26.  A  telegraph  line  441  miles  long  was  constructed  at 
a  cost  of  $443,205.     What  was  the  cost  per  mile  ? 

27.  A  forwarder  ships  93,000  barrels  of  flour  from  St. 
Paul  to  Pittsburgh,  in  cargoes  of  5,136  barrels  each.  How 
many  full  cargoes  does  he  ship  ? 

28.  How  many  bales,  of  396  pounds  each,  can  be  made 
from  84,000  pounds  of  cotton? 

29.  At  what  price  per  head  must  I  sell  105  sheep,  to 
receive  $563.85  ? 

30.  A  stationer  paid  $228  for  gold  pens,  at  $1.90  apiece. 
How  many  pens  did  he  buy  ? 

31.  For  how  many  months  must  I  rent  a  house,  at 
$23.50  per  month,  to  cancel  a  debt  of  $423  ? 

32.  A  chair  maker  received  $172.50  for  chairs,  at  $7.50 
a  set.     How  many  sets  did  he  sell  ? 

33.  The  salary  of  the  President  of  the  United  States  is 
$50,000  a  year.     How  much  is  that  a  day  ? 

^^.  If  the  directors  of  a  railroad  company  appropriate 
$30,000  for  the  purchase  of  passenger  cars,  at  $1,875  each, 
how  many  cars  can  be  bought  with  the  appropriation  ? 


90  SECOND    BOOK  IN  ARITHMETIC. 

35.  1,405,169 ^2,376= how  many? 

36.  One  year  a  manufacturer  received  $2,009  for  gloves, 
at  the  rate  of  $12.25  per  dozen  pairs.  How  many  dozen 
pairs  did  he  sell  ? 

37.  A  grocer  paid  $5,166  for  sugar,  at  $30.75  per  box. 
How  many  boxes  did  he  buy  ? 

38.  How  many  steamboat  bells,  each  weighing  649 
pounds,  can  be  made  from  15,576  pounds  of  bell-metal  ? 

39.  Divide  919,734,140  by  22,705. 

Jfi.  What  is  the  quotient  of  18,382,959-^56J217? 

Case  II.  The  divisor  ending  with  a  cipher  or  ciphers* 
Oral  Work. 

What  is  the  quotient 

7.  Of  50  divided  by  10? 


100.  A.  How  much  is 

1.  i^jj-  of  50  figures  ? 

2.  -^  of  500  figures  ? 
8.  y^  of  500  figures  ? 
^  yVof  250  letters? 

5.  ^of  2,500  letters? 

6.  yfo  of  2,500  letters? 


8.  Of  500  divided  by  10? 

9.  Of  500  divided  by  100? 

10.  Of  250  divided  by  10? 

11.  Of  2,500  divided  by  10? 

12.  Of  2,500  divided  by  100? 


J5.  1.  In  the  number  25,000,  what  value  is  expressed 

by  the  5  ? 

What  vakie  will  be  expressed  by  the  5, 

2.  If  it  be  removed  one  place  to  the  right  ? 

3.  If  it  be  removed  two  places  to  the  right  ? 
Jf.  If  it  be  removed  three  places  to  the  right  ? 

Note.— Ask  questions  similar  to  the  last  four  concerning  the  2  in  this 
number,  then  concerning  the  25. 

How  is  the  number  25,000  affected 

6.  By  removing  one  cipher  from  the  right  ? 

6.  By  removing  two  ciphers  from  the  right  ? 

7.  By  removing  three  ciphers  from  the  right  ? 


INTEGERS.— DIVISION. 


91 


PRmciPLE  II.  Each  removal  of  a  number  one  place 
to  the  right  divides  the  number  by  10. 
101.  A.  What  is  the  quotient 


1.  Of  $.50  divided  by  $.10? 

2.  Of  $5  divided  by  $.10? 

3.  Of  $5  divided  by  100  cents? 
How  much  is 

7.  -JL-  of  $.50,  or  50  cents? 
^  of  $5,  or  500  cents? 
Tk-of$5? 


^.  Of  $2.50  divided  by  $.10? 

5.  Of  $25  divided  by  $.10? 

6.  Of  $25  divided  by  100  cents? 


9. 


10.  -^  of  $2.50,  or  250  cents? 

11.  ^of  $25? 

12.  y^o  of  $25? 

a.  To  change  dollars  to  cents : — Multiply  by  100. 
h.  To  change  dollars  and  cents  to  cents : —  Omit  the  dollar 
mark  and  the  decimal  point. 
What  is  the  quotient  of 


19.  $7.20-4-10? 

20.  $72.50-4-10? 

21.  $725,404-10? 


IS.  $7-4-10?  16.  $74-100? 

U.  $72-r-10?  17.  $72-4-100? 

15.  $7254-10?  18.  $725-^100? 

a.  To  divide  a  number  expressing  dollars  by  10 : — Place  a 

decimal  point  before  the  ones. 
h.  To  divide  a  number  expressing  dollars  by  100  : — Place  a 
decimal  point  before  the  tens. 

J5.  1,  A  farmer  sowed  30  bushels  of  plaster  on  a  field 
of  10  acres.     How  many  bushels  did  he  sow  to  the  acre  ? 

2.  In  a  regiment  are  900  soldiers,  in  companies  of  100 
each.     How  many  companies  are  in  the  regiment  ? 

S.  How  much  is  1  tenth  of  720  boxes  of  figs  ? 


i  J+.  $40  by  10. 
Divide-]  5.  $27  by  10. 
(  6.  $150  by  10. 


7.  $21.50  by  10.       10.  $275  by  100. 

8.  $15.80  by  10.       11.  |830  by  100. 

9.  $134.20  by  10.      12.  $506  by  100. 
IS.  A  wheelwright  received  $60  for  wagon  wheels,  at 

a  set.     How  many  sets  did  he  sell  ? 
i^.  A  housekeeper  paid  one  dollar  for  strawberries,  at 
10  cents  a  quart.     How  many  quarts  did  she  buy  ? 


92  SECOND    BOOK  IN  ARITHMETIC. 

15.  I  have  84  lionrs'  work  to  do.  If  I  work  10  hours 
a  day,  how  many  full  days  shall  I  work,  and  how  many 
hours  shall  I  work  the  last  day  ? 

16.  At  $  .10  apiece,  how  many  pine-apples  can  I  buy 
for  %  .47,  and  how  much  money  will  I  have  left  ? 

17.  $1.50  for  10  melons  is  how  much  for  1  melon? 

18.  $15  for  10  days'  work  is  how  much  a  day  ? 

19.  $84  for  100  straw  hats  is  how  much  for  one  hat  ? 
W.  $6,250  for  100  acres  of  land  is  how  much  an  acre  ? 
^1.  A  plasterer  earned  $36.50  in  10  days.     How  much 

was  that  per  day  ? 

Written  Work. 

103.  Ex.  Divide  2,764  by  10,  and  by  100. 
Explanation. — I  Processes. 

S;\So^       ^.76J,^10  =  276\J,  =  276,^ 

one  figure  from         ^  J 6  J,. -r- 100  =  27  \  6  Jf.  =  2  7  j%\ 

the  right;  and  by 

100,  by  cutting  off  two  figures  from  the  right. 
The  figures  of  the  dividend  not  cut  off  express  the  integers  of 

the  quotient,  and  those  cut  off  express  the  part  of  the  dividend 

remaining  undivided.    This  remainder,  written  over  the  divisor, 

expresses  the  fractional  part  of  the  quotient. 
The  results,  276^^  and  27^*^,  are  the  quotients  required. 

What  is  the  divisor,  what  figures  express  the  integers  of 
the  quotient,  and  what  figures  the  fractional  part, 

1.  When  you  cut  off  one  figure  from  the  right  of  a 
number  ? 

2.  When  you  cut  off  two  figures  from  the  right  ? 

3.  When  you  cut  off  three  figures  from  the  right  ? 

Jf..  When  you  cut  off  four  figures  ?    Five  figures  ?    Six 
figures  ? 

6.  When  you  cut  off  any  number  of  figures  ? 


Divide 

1.  647,500  by  100. 

2.  1,627,000  by  1,000. 

3.  76,275  by  1,000. 


INTEGERS.—DIVISIOS. 
Peoblems. 


93 


Jf.  324,700,000  by  10,000. 

5.  725,000,000  by  100,000. 

6,  32,967,816  by  10,000. 


7.   10  cost  143. 


10.  100  cost  $375. 

11.  100  cost  $1,850. 

12.  1,000  cost  1 


™*^\P^^^^^<?.  10cost$.80. 
Of  1,  When     i  p.  ,0  cost  $262.50. 

13.  A  farmer  having  $1,807,  bought  horses  at  $100 
each.  How  many  horses  did  he  buy,  and  how  many  dol- 
lars had  he  left  ? 

llj..  A  capitalist  invests  $375,000  in  United  States  Gov- 
ernment bonds,  at  $1,000  each.  How  many  bonds  does 
he  buy  ? 

15.  How  many  freight-cars  will  be  required  to  transport 
58,293  barrels  of  flour,  if  100  barrels  make  one  car  load  2 


Oral  Work.- 

1.  ^  of  60  ? 
4-  l^o  of  70? 
7.  yio  of  2,400? 

10.  y^o  of  2,485  ? 
IS.  y^  of  2,785  ? 


B.  What  is 
the  quotient 


6.  ^  of  70  ? 
9.  ^i^- of  2,400? 
12.  ^lo  of  2,485? 


15. 


of  2,785? 


-103.  A.  How  much  is 

2.  i  of  ^  of  60  ?  8.  ^V  of  60  ? 

5.  ^  of  -j\  of  70  ? 
^.  i  of  ^  of  2,400? 
^^'  iof^lo  of  2,485? 
U.  iofyio  of  2,785? 
1.   Of  i^o  of  60-^3? 
3.  Of  ^of  80^3? 
5.  Of  yVof  84^3? 
7.  Of  y^  of  4,800 -r- 8? 
9.  Of  T^o  of  4,837^8? 
Jl.  Of  i^of  5,137-f-8? 

13.  What  is  the  fractional  part  of  the  quotient  in  prob- 
lem 4,  and  how  is  it  formed  ? 


2.  Of  60-f-30? 

4.  Of  80-^30? 

6.  Of  84-f-30? 

8.  Of  4,800^800? 
10.  Of  4,837  —  800? 
12.  Of  5,137-^800? 


Note. — Ask  similar  questions  concerning  problenis  6,  10,  and  12. 


^^  SECOND    BOOK  IN  ARITHMETIC. 

<i  1,  A  milkman  paid  $320  for  cows,  at  $40  apiece. 
How  many  cows  did  he  buy  ? 

2,  At  $50  a  light,  how  many  lights  of  French  plate 
window-glass  can  be  bought  for  $400  ? 

3.  At  $  .60  a  bushel,  how  many  bushels  of  potatoes  can 
be  bought  for  $15  ? 

4-.  A  merchant  tailor  sold  50  coats  for  $450.  How 
much  did  he  receive  apiece  for  them  ? 

6.  A  milliner  received  $7.50  for  ribbon,  at  $  .30  a  yard. 
How  many  yards  did  she  sell  ? 

6,  At  $  .20  apiece,  how  many  water-melons  can  I  buy 
for  $1.38,  and  how  much  money  will  I  have  left  ? 

7,  $720  for  300  yards  of  cloth  is  how  much  a  yard  ? 

8,  A  fruit  dealer  packed  2,250  bushels  of  apples  in  900 
barrels.     How  many  bushels  did  he  put  in  a  barrel  ? 

9,  How  much  is  1  fortieth  of  2,800  barrels  of  flour? 

10.  1,600  pounds  of  pork  are  how  many  barrels  of  200 
pounds  each  ? 

11,  5,400  pounds  of  wheat  are  how  many  bushels  of  60 
pounds  each  ? 

Written  Work.— 104:.  Ex.  Divide  53,485  by  700. 

Explanation. — Since  700  =  7  times 
100, 1  first  divide  by  100,  and  the  Pkocess 

result  thus  obtained  by  7. 

Dividing  53,485  by  100,  I  have  534,  7\00  )53,4.\8S 

with  85  remaining  undivided.  /v  ^  2,8.5. 

Dividing  534,  the  integral  part  of  the  "^  ^ 

first  quotient,  by  7, 1  have  76,  with 

2  remaining  undivided.  This  2  is  a  part  of  the  4  hundred  of  the 
given  dividend,  and  therefore  is  2  hundred.  Annexing  the  first 
remainder,  85,  to  this  2  hundred,  I  have  285,  the  whole  remain- 
der ;  and  this,  written  over  the  given  divisor,  700,  forms  f ff ,  the 
fractional  part  of  the  quotient. 

The  final  result,  76fff ,  is  the  quotient  required. 


INTEGERS.— DIVISION.  95 

Peoblems. 

1.  At  $1,200  each,  how  many  steam-tugs  can  be  bought 
for  $18,000? 

^.  A  lumberman  received  $3,840  for  lumber,  at  $20 
per  thousand  feet.     How  many  thousand  feet  did  he  sell  ? 

3.  A  company  purchase  a  railroad  for  $1,656,750,  and 
the  payments  are  to  be  $250,000  annually.  How  many 
^  payments  do  they  make,  and  how  much  is  the  last  pay- 
mont  ? 

4-.  A  miller  purchased  9,478  pounds  of  wheat.  How 
many  bushels  did  he  buy,  allowing  60  pounds  to  the  bushel  ? 

5.  In  one  ream  of  paper  there  are  20  quires.  How 
many  full  reams  are  there  in  1,976  quires  ? 

6,  An  agent  sold  parlor  organs  at  $130  each,  and  re- 
ceived $6,240.     How  many  organs  did  he  sell  ? 

7.  A  wholesale  grocer  bought  3,440  pounds  of  tea,  in 
80-pound  chests.     How  many  chests  did  he  buy  ? 

8,  There  are  60  minutes  in  an  hour.  1,440  minutes 
are  how  many  hours  ? 

Divide  What  is  the  quotient  of 


9.  387,695  by  4,500. 

10.  382,775  by  2,500. 

11.  8,329,659  by  365,000. 

12.  255,837,432,000  by  700,000. 


13.   299,392-^24,000? 
U.   87,693,275-^41,700? 

15.  10,735,000-^75,000? 

16.  12,600,000-^120,000? 


17.  n  I  pay  $13,750  for  a  farm  of  550  acres,  how  much 
do  I  pay  per  acre  ? 

18.  The  dividend  is  87,693,275,  and  the  divisor  is  41,700. 
Find  the  quotient  and  the  remainder. 

19.  A  Government  agent  bought  cavalry  horses  at  $160 
apiece,  and  expended  $60,160.  How  many  horses  did  he 
buy? 


96  SECOND   BOOK  IN  ARITHMETIC. 

10«l.    EULES   FOR   DlYISION   OF    INTEGERS. 

I.  The  divisor  ending  with  a  digit. 

1.  Write  the  divisor  at  the  left  of  the  dividend^  and 
dra/w  a  line  between  them;  also  a  litie  either  under  or  at 
the  right  of  the  dividend^  to  sejparate  it  from  the  quotient, 

2.  Divide  the  first  jpa/rtial  dividend  hy  the  divisor^  and 
place  the  result  in  the  quotient;  multijply  the  divisor  hy 
this  quotient^  a/nd  subtract  tJie  jproduct  from  the  partial 
dividend. 

3.  Annex  to  the  remainder  the  units  expressed  by  the 
next  figure  of  the  dividend,  and  divide  the  partial  divi- 
dend thus  formed  as  before. 

4.  Proceed  in  the  same  manner  with  each  partial  divi- 
dend; and  J  in  the  quotient,  place  thefi/nal  remainder  over 
the  divisor. 

II.  The  divisor  ending  with  a  cipher  or  ciphers. 

1.  Cut  off  the  cipher  or  ciphers,  and  an  equal  number 
of  figures  from  the  right  of  the  dividend. 

2.  Divide  the  remaining  part  of  the  dividend  by  the 
remaining  part  of  the  divisor,  and  to  the  last  remainder 
annex  the  figures  cut  off  from  the  dividend,  for  the  final 
remainder. 

u.  When  any  partial  dimdend  is  less  than  the  divisor,  place  0  in  the 
quotient;  then,  to  this  partial  dividend  annex  the  units  expressed  hy 
the  next  figure  of  the  dividend,  to  form  the  next  partial  dividend. 
b,  WJien  the  divisor  is  10,  100,  1,000,  etc.,  perform  the  process  men- 
tally. 
Note.— Show  pupils  how  to  proceed 

1.  When  any  product  is  greater  than  the  partial  dividend  from  which  it 
is  to  be  taken, 

2.  When  the  remainder  is  equal  to,  or  greater  than  the  divisor. 

3.  To  find  a  quotient  figure  hij  trial,  when  the  divisor  is  expressed  by 
more  than  one  figure;  i.e.,  to  use  the  first  one  or  two  left-hand  figures 
of  the  divisor  for  a  trial  divisor,  and  an  equal  number,  or  one  more,  of  the 
left-hand  figures  of  the  dividend  for  a  trial  dividend. 


INTEGERS.— DIVISION, 


97 


Peoblems. 
A.   I  ^  1  ±  5 

6)4,986     8)7,300,528     29)70,235(     10)87,690    100)37,296 

6  11 

2,500)998,600(        500,000)725,000,000  705)806,450( 


468)908,808( 


10.  Divide  4,368  by  6,  and  by  24. 


Making  the  Computations. 

6  in  43,  7  times  (1  remaining) ;  write  7. 
6  in  16,  2  times  (4  remaining);  write  2. 
6  in  48,  8  times;  writes.  Result,  728. 

24  in  43,  once;  write  1.     Once  24  is  24;  24 

from  43  leaves  19 ;  annex  6. 
24  in  196,  8  times;  write  8.     8  times  24  are 

192;  192  from  196  leaves  4;  annex  8. 
24  in  48,  2  times;  write  2.    2  times  24  are  48; 

48  from  48  leaves  0. 
Result,  182. 


Divide 

11,   3,279  by  6 ;  by  8 ;  by  7. 
1^,   41,220  by  10 ;  by  15 ;  by  20. 

13.   $408.75  by  $3.27;  by  $1.25. 

U.   1,050,  22,365,  and  50,000  by  700. 

15.  $25.25,  $100.75,  and  $492.50  by  25 

16.  996,  48,009,  and  65,525  by  150. 
21.  493-T-12  = 


Pkooessbs. 

6)4., 368 

728 

2 

)4., 368(18 

2i 
196 

19^ 

48 
48 

Dividends.  Divisors.  Quotients. 

17.  1,458  9  — 

18.  $763  5  — 

19.  1,728  6  — 
^20.  $8.64  $.08  — 


22.  518—32  = 

23.  674-r-13  = 
2Jf.  $72.60-^15r= 
25.  $5,500-r-$1.25  = 

E 


26. 


27. 


8,654,300__ 
9,000     "~ 

7,000,888_ 
758       "" 

4,235,262  _ 
1,294     "" 


98  SECOND   BOOK  IN  ARITHMETIC, 

Oral  Work. — 1.  A  furrier  received  $400  for  bufEalo- 
robes,  at  $10  apiece.     How  many  robes  did  lie  sell  ? 

^.  Carrie  paid  $  .90  for  needles,  at  $  .06  a  paper.  How 
many  papers  did  she  buy  ? 

3.  $588  for  6  lumber  wagons  is  how  much  apiece  ? 

Jf,,  $475  for  5  acres  of  land  is  how  much  per  acre  ? 

5.  $3.76  for  4  baskets  of  peaches  is  how  much  a  basket  ? 

6.  A  lumberman  banked  110  logs  in  5  days.  How 
many  logs  was  that  per  day  ? 

7.  If  a  Mississippi  steamboat  bums  7  cords  of  wood 
per  day,  in  how  many  days  will  she  bum  301  cords  ? 

8.  How  many  hours  will  it  take  a  steamboat  to  run 
270  miles,  running  9  miles  an  hour  ? 

9.  A  joiner  cut  a  strip  of  moulding  168  inches  long, 
into  7  equal  pieces.     How  long  was  each  piece  ? 

10,  At  $  .10  apiece,  how  many  copy-books  can  I  buy 
with  $  .75,  and  how  much  money  will  1  have  remaining  ? 

Written  Work. — JB.  1,  How  many  sheep  are  1  fourth 
of  16,236  sheep  ? 

2,  An  Ohio  farmer  exchanged  his  wheat  crop  of  1,965 
bushels,  for  flour,  receiving  1  barrel  of  flour  for  every  5 
bushels  of  wheat.     How  much  flour  did  he  receive  ? 

3.  If  a  ship  sails  from  New  York  to  Greece — 4,800 
miles — in  25  days,  .what  is  her  daily  rate  of  speed  ? 

^.  A  company  of  96  immigrants  purchased  a  tract  of 
43,200  acres  of  Texas  lands,  which  they  shared  equally. 
How  many  acres  did  each  immigrant  receive  ? 

6.  How  many  dress  patterns  of  13  yards  each,  can  be 
cut  from  a  piece  of  mohair  containing  43  yards  ? 

6.  A  dealer  received  $1,680  for  sewing-machines,  at 
$35  apiece.     How  many  machines  did  he  sell  ? 

7.  In  how  many  days  can  112  men  do  4,928  days'  work  ? 


INTEGERS.— DIVISION.  99 

S,  A  fruit  dealer  sold  686  baskets  of  peaches  for 
$1,543.50.     What  was  the  price  per  basket  ? 

9.  A  nurserj-man  sold  6,872  young  apple-trees  for 
$1,030.80.     How  much  did  he  receive  apiece  for  them? 

10.  1  hundredth  of  50,000  oranges  are  how  many  oranges  ? 

11.  A  pork  buyer  packed  237,600  pounds  of  pork  in 
barrels  of  200  pounds  each.    How  many  barrels  did  he  fill  ? 

1^.  If  a  factory  makes  275  yards  of  cloth  daily,  in  how 
many  days  will  it  make  57,475  yards  ? 

13.  The  expenses  of  a  party  of  8  men  on  a  journey  to 
California,  were  $1,072.     What  was  each  man's  share  ? 

14..  A  farmer  harvested  2,520  bushels  of  oats  from  36 
acres.     What  was  the  yield  per  acre  ? 

15.  A  forwarder  shipped  74,232  bushels  of  grain  in  18 
equal  cargoes.     How  many  bushels  were  in  each  cargo  ? 

16.  A  farmer  made  962  gallons  of  cider,  which  he  put 
into  casks  holding  41  gallons  each.  How  many  full  casks 
had  he? 

17.  An  army  contractor  paid  $39,865  for  2,345  barrels 
of  beef.     How  much  did  the  beef  cost  him  per  barrel  ? 

18.  A  miller  purchased  1,157  bushels  of  wheat  weigh- 
ing 69,420  pounds.     What  was  the  weight  of  one  bushel  ? 

19.  The  yearly  cost  of  keeping  a  turnpike-road  in  repair 
is  $1,407.60,  at  the  rate  of  $30.60  per  mile.  How  many 
miles  long  is  the  road  ? 

W.  39,520  quires  of  paper  are  1,976  reams.  How  many 
quires  are  one  ream  ? 

21.  How  many  kettles,  each  weighing  348  pounds,  can 
be  made  from  20,000  pounds  of  iron  ? 

106.  The  average  of  two  numbers  is  one  half  of  their 
sum ;  of  three  numbers,  one  third  of  their  sum ;  of  four 
numbers,  one  fourth  of  their  sum ;  and,  in  general, 


100  SECOND   BOOK  IN  ARITHMETIC, 

The  average  of  two  or  more  numbers  is  the 

quotient  of  their  sum  divided  by  the  number  of  numbers. 

Pkoblems. 
What  is  the  average 


Of  284  and  150? 
Of  375  and  645? 
Of  49,  196,  and  442? 


Jf.  Of  266,  55,  and  738? 

5.  Of  84,  79,  263,  and  590? 

6,  Of  808,  103,  and  643? 


7.  The  ages  of  3  boys  are  6,  9,  and  12  years.  What  is 
the  average  of  their  ages  ? 

8.  What  is  the  average  width  of  a  board  that  is  8 
inches  wide  at  one  end  and  14  inches  wide  at  the  other  ? 

9.  A  man  walked  in  3  successive  days  47  miles,  29 
miles,  and  17  miles.    What  was  his  average  daily  distance  ? 

10.  A  grocer  bought  3  hogsheads  of  molasses  contain- 
ing, respectively,  135  gallons,  143  gallons,  and  127  gallons. 
What  was  the  average  number  of  gallons  to  a  hogshead  ? 

11.  In  a  village  school  the  number  of  pupils  in  attend- 
ance Monday  was  134,  Tuesday  128,  Wednesday  143, 
Thursday  133,  and  Friday  147.  What  was  the  average 
daily  attendance  ? 

12.  At  a  carpet  manufactory  19,110  yards  of  carpet 
were  woven  in  78  days.  What  was  the  average  number 
of  yards  woven  daily  ? 

13.  Find  the  average  price  of  28  acres  of  land  at  $36 
an  acre,  and  35  acres  at  $27  an  acre. 

IJf,.  A  merchant's  sales  Monday  amounted  to  $348.91, 
Tuesday  to  $317.07,  Wednesday  to  $294.63,  Thursday  to 
$336,  Friday  to  $332.87,  and  Saturday  to  $369.  How 
much  were  his  average  daily  sales  ? 

Note  1. — For  outlines  of  division  for  review,  see  pa^e  370. 

Note  2. — Teachers  who  prefer  that  decimals  should  be  studied  in  con- 
nection with  integers,  will  now  require  tlieir  pupils  to  study  division  of 
decimals,  pages  130-138. 


SECTION  tL^ 

SIXTEEN  GENERAL  PROBLEMS  IN  INTEGERS. 

Oral  Work.— 107.  A.  L  The  parts  are  21,  7,  5, 
and  9.     What  is  the  sum? 

^.  Charles  has  17  cents,  James  has  13  cents,  John  has 
25  cents,  and  Luke  has  20  cents.  What  sum  of  money 
have  the  four  boys  ? 

S.  The  sum  of  two  numbers  is  48,  and  one  of  the  num- 
bers is  18.     What  is  the  other  number? 

^.  In  two  nests  are  27  eggs,  and  9  of  them  are  in  one 
nest.     How  many  eggs  are  in  the  other  nest  ? 

5.  The  sum  of  five  numbers  is  56,  and  four  of  the  num- 
bers are  12,  7,  8,  and  11.     What  is  the  other  number? 

6.  1  have  a  farm  of  80  acres,  divided  into  seven  fields. 
Six  of  the  fields  contain,  respectively,  15  acres,  10  acres, 
13  acres,  7  acres,  18  acres,  and  9  acres.  How  many  acres 
does  the  seventh  field  contain  ? 

Note.  —  Require  pupils  to  state,  briefly  and  accurately,  the  process 
called  for  by  each  of  the  sixteen  problems  in  this  section. 

Problem  I.  The  parts  given  ;  to  find  their  sum. 
Problem  II.  The  sum  of  two  numbers^  and  one  of 

them  given  'j   to  find  the  other  number. 

Problem  III.  The  sum  of  more  than  two  numbers^  and 

all  but  one  of  them  given  ^   to  fi/rid  that  one, 

JB.  1.  What  is  the  difference  between  75  and  89  ? 

2.  The  greater  of  two  numbers  is  51,  and  the  less  is 
30.     What  is  the  difference  ? 

S.  The  foremast  of  a  ship  is  83  feet  high,  and  the  main- 
mast is  99  feet  high.  What  is  the  difference  in  their 
heights  ? 


1^2  SRqOJ^l^   BOOK  IN  ARITHMETIC. 

Ju.  .The  jniuuend  is  42,  and  the  subtrahend  is  25.  What 
if5 ^ii53,-rbln3i{litidier/^  :*\  -J 

5.  I  have  36  bushels  of  apples.  If  I  keep  22  bushels, 
and  sell  the  remainder,  how  many  bushels  shall  I  sell  ? 

6.  The  minuend  is  42,  and  the  remainder  is  17.  What 
is  the  subtrahend  ? 

7.  A  butcher  having  $63,  paid  a  part  of  his  money  for 
sheep,  and  had  $18  remaining.  How  much  did  the  sheep 
cost  him  ? 

8.  The  greater  of  two  numbers  is  89,  and  the  differ- 
ence is  55.     What  is  the  less  number  ? 

9.  A,  who  has  more  money  than  B,  has  $125  ;  and  the 
difference  between  his  money  and  B's  is  $82.  How  much 
money  has  B  ? 

10,  The  subtrahend  is  32,  and  the  remainder  is  68. 
What  is  the  minuend? 

11,  How  many  chestnuts  must  a  boy  have,  that  he  may 
eat  75  of  them,  and  have  a  remainder  of  45  ? 

12,  The  less  of  two  numbers  is  130,  and  their  difference 
is  95.     What  is  the  greater  number  ? 

IS.  In  the  less  of  two  piles  of  wood  are  135  cords,  and 
the  difference  between  the  piles  is  45  cords.  How  many 
cords  of  wood  are  in  the  greater  pile  ? 

Problem  IV.  Two  numbers  given;  to  find  their  differ- 
ence. 

Problem  Y.  The  minuend  and  subtrahend  given;  to 
find  the  remainder. 

Problem  YI.  The  minuend  and  remainder  given;  to 
find  the  subtrahend. 

Problem  YII.  The  subtrahend  and  remainder  given; 
to  find  the  minuend. 


GENERAL   PROBLEMS   IN  INTEGERS.        103 

Peoblem  VIII.  The  greater  of  two  nurribersj  and  their 
difference  given;  to  find  the  less  number. 

Peoblem  IX.  The  less  of  two  numbers^  and  their  dif- 
ference given  ;  to  find  the  greater  number. 

C.  1.  The  multiplicand  is  16,  and  the  multiplier  is  7. 
What  is  the  product  % 

'2.  The  factors  are  %  8,  5,  and  4.   What  is  the  product  ? 

S.  In  8  windows  of  12  panes  of  glass  each,  are  how 
many  panes  of  glass  ? 

^.  At  3  cents  each,  how  much  will  1  dozen  lemons 
cost  %    How  much  will  5  dozen  cost  ? 

5.  The  product  is  72,  and  the  multiplier  is  4.  What 
is  the  multiplicand  ? 

6.  The  product  is  96,  and  the  multiplicand  is  16. 
What  is  the  multiplier  ? 

7.  I  paid  $72  for  4  tons  of  hay.  What  was  the  price 
of  1  ton? 

^.96  tons  of  freight  will  make  how  many  car  loads,  of 
16  tons  each? 

9.  The  product  of  four  factors  is  320,  and  three  of  the 
factors  are  2,  5,  and  4.     What  is  the  other  factor  ? 

10.  I  pay  5  carpenters  $180,  at  the  rate  of  $2  a  day. 
How  many  weeks  do  they  work  ? 

Peoblem  X.  The  multijplicand  and  multijplier  given; 
to  find  the  jproduct. 

Peoblem  XI.  Any  number  of  factors  given;  to  find 
the  ^product. 

Peoblem  XII.  The  product  of  two  factors^  and  either 
factor  given;  to  find  the  other  factor. 

Peoblem  XIII.  The  product  of  m^ore  than  two  factors^ 
and  all  the  factors  but  one  given  ;  to  fi/ifid  that  factor. 


104  SECOND   BOOK  IN  ARITHMETIC. 

2>.  1.  The  dividend  is  144,  and  the  divisor  is  8.  What 
is  the  quotient  ? 

^.  360  bushels  of  oats  will  fill  how  many  3-bushel  bags  ? 

c^.  96  panes  of  glass  for  8  windows,  are  how  many- 
panes  for  1  window  ? 

4-.  The  divisor  is  7,  and  the  quotient  is  14.  What  is 
the  dividend  ? 

5.  The  divisor  is  8,  and  the  quotient  is  6f .  What  is 
the  dividend  ? 

What  is  the  dividend 

6.  If  the  divisor  is  8  miles,  and  the  quotient  is  15  ? 

7.  If  the  divisor  is  12  dozen,  and  the  quotient  is  9^^? 
How  much  money  must  I  have,  that  I  may  divide  it 

8.  Among  16  persons,  and  give  each  person  $10  ? 

9.  Among  16  persons,  and  give  each  person  $10|^? 

10.  The  dividend  is  $2.25,  and  the  quotient  is  25.  What 
is  the  divisor  ? 

11.  The  dividend  is  147  hours,  and  the  quotient  is  21 
hours.    What  is  the  divisor  ? 

Problem  XIV.  The  divisor  and  dividend  given;  to 
jmd  the  quotient. 

Problem  XV.  The  divisor  and  quotient  given  ;  to  find 
the  dividend. 

Problem  XVI.  The  dividend  and  quotient  given;  to 
find  the  divisor. 

Written  Work.  —  l.  The  parts  are  954,   3,420,  75, 

3,659,    251,    8,000,   11,756,   8,   9,999,   730,   and  28,028. 
What  is  the  sum? 

2.  The  minuend  is  9,000,001,  and  the  subtrahend  is 
365,497.     What  is  the  difference  ? 

3.  Take  $324.09  from  $1,000.    What  is  the  remainder  ? 


GENERAL  PROBLEMS   IN  INTEGERS.  105 

^.  The  multiplicand  is  72,394,  and  the  multiplier  is 
3,600.     What  is  the  product  ? 

6.  The  sum  of  two  numbers  is  31,654,  and  one  of  the 
numbers  is  2,870.     What  is  the  other  number  ? 

6.  What  number  is  that,  the  factors  of  which  are  93,268 
and  50  ? 

7.  The  quotient  is  94,  and  the  divisor  is  317.  What 
is  the  dividend  ? 

8.  The  divisor  is  $25.50,  and  the  dividend  is  $1,810.50. 
What  is  the  quotient  ? 

9.  8,400  +  9 +  55,374+ 286  + 6,857 +2,002+ how  many 
=  100,000? 

10,  $25,408~$18,976=how  many  times  $24? 

11.  36  X  25  X  300  x  46  is  50  times  what  number  ? 

12,  The  minuend  is  875,004,  and  the  remainder  208,365. 
What  is  the  subtrahend  ? 

13.  From  what  number  must  $964.87  be  taken,  to  leave 
$215.13? 

i^.  By  what  number  must  23,040  be  divided,  to  ob- 
tain 36  ? 

15.  Dividing  a  certain  number  by  70,000, 1  obtain  Q%  for 
the  integer  of  the  quotient,  and  60,000  for  a  final  remain- 
der.    What  is  the  number  divided  ? 

16,  100 +  67 +  229 +  80 +  924=: I  of  what  number? 
i7.  527  X  55  X  9  -  253,974  =  3  times  how  much  ? 

18.  17,984^281  =  28 +how  many? 

19.  809  times  the  quotient  of  $525.75  ^$21.03  is  how 
much  more  than  5,000  ? 

20.  What  number  is  that  to  which  if  267  be  added,  from 
the  sum  438  be  subtracted,  the  remainder  be  multiplied  by 
6,  and  the  product  be  divided  by  12,  the  quotient  will  be 
491? 

E2 


SECTION  VII. 

GENERAL  REVIEW  PROBLEMS  IN  INTEGERS. 

Oral  Work. — 1.  How  mucli  can  a  boy  earn  in  5  days, 

at  $  .31  a  day  ? 

2.  Find  the  cost  of  10  yards  of  calico,  at  13  cents  a 
yard ;  and  8  yards  of  ribbon,  at  20  cents  a  yard. 

3.  At  ten  cents  a  pound,  how  much  will  a  grocer  pay 
for  5  barrels  of  crackers  of  41  pounds  each  ? 

^.  A  fruiterer  bought  48  boxes  of  lemons,  at  $2  a  box, 
and  sold  them  all  for  $125.     How  much  did  he  gain  ? 

5.  How  many  pews  are  there  in  a  church  that  seats 
300  persons,  5  persons  in  a  pew  ? 

6.  If  I  owe  $54,  $21,  and  $15,  and  I  pay  all  but  $25, 
how  much  do  I  pay  ? 

7.  If  5  cords  of  wood  are  worth  15  dollars,  how  much 
are  7  cords  worth  ? 

S.  A  butcher  killed  8  sheep,  the  total  weight  of  which 
was  720  pounds.     What  was  their  average  weight  ? 

9.  A  dealer  sells  melons  at  10  cents,  13  cents,  15  cents, 
and  18  cents  apiece,  according  to  size.  What  is  the  average 
price  ? 

10,  If  the  attendance  at  school  Monday  is  38,  Tuesday 
43,  Wednesday  39,  Thursday  40,  and  Friday  45,  what  is 
the  average  attendance  for  the  week  ? 

Written  Work. — 1.  A  salary  of  $1,000  per  year  is 
how  much  per  month  ?     How  much  per  week  ? 

2.  A  grain  buyer  receives  an  order  for  12,500  bushels 
of  wheat,  and  he  has  only  7,645  bushels  in  store.  How 
many  bushels  must  he  purchase  to  fill  the  order  ? 

S.  At  $6  a  week  for  board,  how  much  are  the  receipts 
from  45  boarders  in  26  weeks  ? 


EUVIIJW  PROBLEMS  IN  INTEGERS.  107 

^.  One  week  a  fruit  dealer  bought  123  barrels,  1,075 
barrels,  3,550  barrels,  1,805  barrels,  987  barrels,  and  562  bar- 
rels of  apples.     How  many  barrels  of  apples  did  he  buy  ? 

6,  A  ship  and  cargo  are  valued  at  $130,480.50,  and  the 
cargo  is  worth  $55,297.50.    What  is  the  value  of  the  ship  ? 

6.  How  many  bales  will  1,072,512  pounds  of  cotton 
make,  allowing  400  pounds  to  the  bale  ? 

7.  If  I  travel  west  from  Philadelphia  8  hours,  at  the 
rate  of  34  miles  an  hour,  and  then  travel  east  176  miles, 
how  f a-r  will  I  be  from  Philadelphia  ? 

8.  In  building  a  bam  I  used  7  thousand  feet  of  lumber, 
that  cost  me  $15  a  thousand ;  and  I  paid  $139  for  other 
materials  and  labor.     How  much  did  the  barn  cost  me  ? 

9.  In  a  factory  432,740  yards  of  cassimere  were  made 
in  308  days.     How  many  yards  were  made  daily  ? 

10.  In  the  manufacture  of  this  cloth  616,000  pounds  of 
wool  were  used.     How  many  pounds  were  used  daily  ? 

11,  If  1,071,399  yards  of  cotton  cloth  are  made  from 
357,133  pounds  of  cotton,  how  many  yards  of  cloth  are 
made  from  one  pound  of  cotton  ? 

m.  Three  men  buy  a  lot  of  land  for  $6,000,  and  build 
a  store  upon  it  at  a  cost  of  $7,281.  How  many  dollars 
does  each  man  pay  ? 

13,  A  coal  dealer  bought  1,050  tons  of  coal,  receiving 
2,240  pounds  for  a  ton,  and  sold  it  at  2,000  pounds  for  a 
ton.     How  many  tons  did  he  sell  ? 

lit..  What  is  the  average  weight  of  8  bales  of  cotton 
which  weigh  respectively  385,  367,  418,  374,  396,  405, 
373,  and  402  pounds  ? 

15.  I  buy  real-estate  for  $16,575,  agreeing  to  pay  for  it 
in  yearly  payments  of  $1,200  each.  How  many  payments 
will  I  make,  and  how  much  will  be  the  last  payment  ? 


108  SECOND    BOOK  IN  ARITHMETIC. 

16.  In  June  a  dairyman  made  355  pounds,  424  pounds, 
391  pounds,  and  330  pounds  of  butter ;  and  packed  it  in 
tubs  of  50  pounds  each.     How  many  tubs  did  he  fill  ? 

Oral  Work. — B.  1.  Ira's  age  is  6  years,  and  Paul's  is 
20.    In  how  many  years  will  Paul  be  twice  as  old  as  Ira  ? 

2.  In  a  certain  school  are  50  boys,  and  4  times  as  many 
girls  lacking  17.     How  many  pupils  are  in  the  school  ? 

3.  If  11  apples  cost  $  .07,  how  many  apples  can  I  buy 
for  $  .28  ? 

^.  If  6  bananas  cost  $  .42,  what  do  15  bananas  cost  ? 

5.  How  many  tons  of  hay  at  6  dollars  a  ton,  will  pay 
for  8  yards  of  cloth  at  $3  a  yard  ? 

6.  A  wood  dealer  sold  to  three  persons  12  cords,  10 
cords,  and  36  cords  of  wood,  and  had  29  cords  left.  How 
many  cords  had  he  at  first  ? 

7.  From  a  stock  of  100  thousand  shingles,  53  thousand 
were  sold  to  one  builder,  and  31  thousand  to  another. 
How  many  shingles  were  unsold  ? 

8.  A  merchant  has  74  yards  of  cloth  in  three  pieces, 
one  of  which  contains  29  yards,  and  another  32  yards. 
How  many  yards  are  there  in  the  third  piece? 

9.  In  how  many  days  will  8  men  do  as  much  work  as 
12  men  can  do  in  10  days  ? 

10.  In  how  many  weeks  will  14  cattle  eat  as  much  as 
6  cattle  will  eat  in  7  weeks  ? 

11.  A  man  has  a  job  of  work  that  he  can  do  in  192 
days.  If  he  employs  5  men  to  assist  him,  in  how  many 
days  will  the  work  be  completed  ? 

12.  If  15  men  can  clear  an  acre  of  ground  in  8  days,  in 
how  many  days  can  12  men  clear  it  ? 

13.  If  a  stage  coach  runs  54  miles  in  9  hours,  in  how 
many  hours  will  it  run  96  miles  ? 


RFVIBW  PROBLEMS  IN  INTEGERS.  109 

Written  Work. — 1.  A  farmer  had  3  flocks  of  sheep, 
the  first  containing  184,  the  second  218,  and  the  third  65. 
He  sold  114  from  the  first  flock,  189  from  the  second,  and 
48  from  the  third.     How  many  sheep  had  he  then  ? 

^.  A  grocer  bought  two  tubs  of  maple  sugar  —  one 
weighing  67  and  the  other  93  pounds — at  $  .15  a  pound, 
and  sold  it  at  $  .18  a  pound.     How  much  did  he  gain? 

3.  I  bought  a  farm  wagon  for  $92,  and  a  family  car- 
riage for  $172 ;  and  paid  for  them  with  l)eef,  at  $12  a 
hundred  pounds.  How  many  hundred  pounds  of  beef 
did  it  take  ? 

4-.  Find  the  cost  of  115  pounds  of  hams,  at  $  .11  a 
pound ;  and  43  pounds  of  bacon,  at  $  .13  a  pound. 

6.  A  fruit  dealer  having  2,468  barrels  of  apples,  sold 
1,382  barrels  for  $4,146,  and  the  remainder  for  $4,344. 
How  much  did  he  receive  per  barrel  for  each  of  the  two 
lots? 

6.  A  farmer  sold  his  farm  of  36  acres  at  $40  per  acre, 
and  invested  the  proceeds  in  Western  lands  at  $2.25  per 
acre.     How  many  acres  did  he  buy  ? 

7.  A  drover  bought  cattle  at  $49.63  per  head,  and  sold 
them  at  $63.50  per  head,  thereby  gaining  $1,511.83.  How 
many  head  of  cattle  did  he  buy  ? 

8.  Last  year  a  bank  teller  received  a  salary  of  $1,250, 
and  his  personal  expenses  were  $753.  He  bought  a  vil- 
lage lot  for  $225,  and  paid  $149  for  improvements  upon 
it.     How  much  of  his  year's  salary  had  he  left  ? 

9.  I  bought  a  stock  of  goods  for  $15,284,  paying  $2,684 
cash,  and  the  balance  in  monthly  payments  of  $1,575  each. 
How  many  monthly  payments  did  I  make  ? 

10.  52  ladies  and  39  gentlemen  went  on  an  excursion, 
and  their  expenses,  which  were  $3  each,  were  paid  by  the 
gentlemen.     How  much  did  each  gentleman  pay  ? 


110  SECOND    BOOK  IN  ARITHMETIC. 

11,  A  young  man  worked  a  year  at  $26  a  month.  He 
paid  $13  a  month  for  board,  and  his  other  expenses  were 
$100.     How  much  money  did  he  save  ? 

12,  A  merchant  bought  16  pieces  of  dress  flannels,  of 
43  yards  each.  After  selling  155  yards,  how  many  dress 
patterns  of  13  yards  each  had  he  ? 

IS.  A  man  contracted  to  deliver  at  a  steamboat  landing 
5,000  cords  of  wood.  He  delivered  76  cords  each  day  for 
35  days,  and  54  cords  each  day  for  24  days.  How  many 
cords  were  yet  to  be  delivered  ? 

llf,,  A  captain,  mate,  and  12  sailors  captured  a  prize  of 
$2,240 ;  the  captain  took  14  shares,  the  mate  6  shares,  and 
each  of  the  sailors  1  share.     How  much  did  each  receive  ? 

15,  A  farm-house  is  worth  $2,450 ;  the  farm  is  worth 
12  times  as  much,  less  $600 ;  and  the  stock  is  worth  twice 
as  much  as  the  house.  How  much  are  house,  stock,  and 
farm  worth  ? 

16,  I  paid  $48  an  acre  for  a  wood  lot  of  60  acres.  I 
sold  the  wood  for  $2,378,  and  the  land  for  $18  an  acre. 
Did  I  gain  or  lose,  and  how  much  ? 

17,  A  clothier  invested  $3,000  in  business.  The  first 
year  he  gained  $756,  and  the  second  year  he  lost  $1,652. 
The  next  three  years  his  average  yearly  gain  was  $748. 
How  much  was  he  worth  at  the  end  of  the  five  years  ? 

18,  Monday  morning  Mr.  D.  had  in  bank  $5,878.50. 
During  the  week  he  drew  checks  for  $360.25,  $145,  and 
$75.50 ;  and  deposited  $1,250,  $750,  $1,000,  and  $1,500. 
What  was  his  balance  in  bank  at  the  close  of  the  week  ? 

19,  A  farmer's  wheat  crop  brought  him  $650,  his  barley 
$205,  his  corn  $97,  and  his  oats  $150.  He  paid  a  farm 
hand  $13  per  month  for  9  months,  paid  $250  for  fertil- 
izers, and  $65.75  for  utensils  and  repairs.  How  much  did 
he  clear  from  his  farm  that  year  ? 


CHAPTER    II. 

DECIMALS. 


SECTION  I. 

NOTATION  AND  NUMERATION. 

108,  A.  One  tenth  is  one  of  the  ten  equal  parts  into 
which  the  unit  one  is  divided. 


ten  tenths. 


A  figure  at  the  right  of  ones  expresses  tenths. 
Tenths  are  written 

1  tenth,  .1       4  tenths,  .4       7  tenths,  .7 

2  tenths,  .2       5  tenths,  .5       8  tenths,  .8 

3  tenths,  .3       6  tenths,  .6       9  tenths,  .9 

JB.  Dividing  each  of  the  10  tenths  of  a  1  into  10  equal 
parts  divides  the  1  into  10  times  10,  or  100  equal  parts. 
One  hundredth  is  1  tenth  of  1  tenth. 


/r/'//  /  /777 


One  tenth         is  ten  hundredths. 

A  figure  at  the  right  of  tenths  expresses  hundredths. 


112 


SECOND   BOOK  IN   ARITHMETIC. 


Hundredths  are  written 


1  hundredth,    .01 

2  hundredths,  .02 

3  hundredths,  .03 


4  hundredths,  .04 

5  hundredths,  .05 

6  hundredths,  .06 


1  hundredths,  .07 

8  hundredths,  .08 

9  hundredths,  .09 


C.  Dividing  each  of  the  100  hundredths  of  a  1  into  10 
equal  parts  divides  the  1  into  10  times  100,  or  1,000  equal 
parts. 

One  thousandth  is  1  tenth  of  1  hundredth. 


One  hundredth      is 


ten  thousandths. 


A  figure  at  the  right  of  hundredths  expresses  thousandths. 
Thousandths  are  written 


1  thousandth,    .001 

2  thousandths,  .002 

3  thousandths,  .003 


4  thousandths,  .004 

5  thousandths,  .005 

6  thousandths,  .006 


7  thousandths,  .007 

8  thousandths,  .008 

9  thousandths,  .009 


109.  A  decimal  unit  is  one  of  the  equal  decimal 

parts  into  which  the  unit  1  is  divided. 

1  tenth,  1  hundredth,  1  thousandth,  1  ten-thousandth,  and  so  on, 
are  decimal  units. 

110.  A  deci^nal  is  a  number  that  expresses  one  or 
more  decimal  units. 

7  tenths,  36  hundredths,  258  thousandths  are  decimals. 

a.  A  number  consisting  of  an  integer  and  a  decimal  is  a  mixed 
nufnber  ;  as  8  and  25  hundredths. 

b.  The  period  or  point  ( . )  placed  before  tenths  is  the  decimal 
"point.    It  is  read  mid.    4.5  is  read  '*4  and  5  tenths." 

111.  Tenths  and  hundredths  are  read  together  as  hun- 
dredths ;  and  tenths,  hundredths,  and  thousandths  are  read 
together  as  thousandths. 


DECIMALS.'-NOTATION  AND   NUMERATION.     113 


Exercises. 
A*  Write  and  read  the  numbers  expressed  by 

1,  8  tenths ;  3  tenths ;  6  tenths ;  2  ones  and  5  tenths. 

^.  1  tenth  and  1  hundredth,  or  11  hundredths. 

S,  3  ones  and  27  hundredths ;  18  and  2  hundredths. 

Jf,,  4  tenths  5  hundredths  and  6  thousandths,  or  456 
thousandths. 

5,  8  tens  3  tenths  and  5  thousandths. 

6,  50  and  16  thousandths ;  87  and  87  thousandths. 

7.  500  and  4  thousandths ;  504  thousandths. 

8.  9,000  and  9  thousandths ;  101,000  and  101  thousandths. 


Write  in  words 

^.  .5;  6.4;  40.6;  .03;  3.27;  .009. 

5.  298.02;  .063;  .175;  19.02. 

6,  80.001;  9.018;  300.300. 


B.  Kead 

1.  .9;  5.2;  .07;  8.05;  50.5. 

2.  .08;  .83;  200.09;  .002;  .901. 
8.  4.023;  6.400;  .365;  500.5. 

C,  Express  by  figures 

i.  Five  tenths ;  nine  hundredths  ;  thirteen  hundredths. 

2.  Forty-eight  hundredths ;  two  thousandths ;  fifty-one 
thousandths. 

3,  Eighty  thousandths ;  six  hundred  five  thousandths. 
^.  Three  and  nine  tenths ;  twenty  -  five  and  fifty  -  two 

hundredths ;  sixty-two  and  ninety-five  thousandths. 

6.  Seven  and  seven  hundredths;  one  and  eleven  hun- 
dredths ;  five  and  three  hundred  seventy-six  thousandths. 

113.  A  decimal  expressed  by 


one  figure 
two  figures 
three  figures 
four  figures 
five  figures 
six  fip'ures 


tenths, 

hundredths, 

thousandths, 

ten-thousandths,- 

hundred-thousandths, 

millionths, 


1  tenth, 

1  hundredth, 

1  thousandth, 

1  ten-thousandth, 

1  hundred-thousandth, 

1  milHonth. 


And  similarly  for  any  greater  number  of  decimal  figures. 


114 


SECOND   BOOK  IN  ARITHMETIC, 


113.  Orders  of  units  at  equal  distances  to  the  right 
and  left  of  ones  have  corresponding  names,  as  shown  in  the 

DIAGRAM   OF   DECIMAL   NOTATION. 

98765-4  32  1.  23466789 


\ 


^\j  #  /  /  #  /  # 

■"     ^''  <s>y  s^^'  ^■''   w   ^/ 


How  many  decimal  figures  are  required  to  express 

1.  Hundredths?     I    S,  Tenths?  5.  Ten-thousandths? 

2.  Thousandths?    |    4.  Millionths?        6.  Ten-raillionths? 
7.  Hundred-thousandths?        S.  Hundred-millionths ? 

What  is  the  name  of  a  decimal  expressed 


9,  By  three  figures  ?  I 
10,  By  five  figures  ?    | 


11.  By  two  figures  ? 

12,  By  one  figure? 


IS.  By  four  figures! 
14.  By  six  figures  ? 


How  many  figures  are  required  to  express 


15.  Tens?     Tenths? 

16.  Hundreds?     Hundredths? 

17.  Millions?     Millionths? 

18.  Thousands?    Thousandths? 

19.  Ten-millions? 

Ten-millionths  ? 


20.  Hundred-thousands? 

Hundred-thousandths  ? 

21.  Hundred-millions? 

Hundred-millionths  ? 

22.  Ten-thousands? 

Ten-thousandths  ? 


"What  place,  counting  from  ones,  is  the  place 


23.  Of  thousands?  SO. 

24.  Of  ten-thousandths?  SI. 

25.  Of  millions?  S2. 

26.  Of  hundredths?  SS. 

27.  Of  thousandths?  SJ^. 

28.  Of  ten-millions  ?  35. 

29.  Of  hundreds?  36. 

37.  Of  millionths? 


Of  hundred-thousandths  ? 

Of  hundred-millions? 

Of  hundred-millionths  ? 

Of  tens  ?     Of  tenths  ? 

Of  ten-thousands  ? 

Of  ten-millionths  ? 

Of  hundred-thousands  ? 


DECIMALS.—NOTATION  AND  NUMERATION.    115 


Of  what  order  of  units,  integral  or  decimal, 


S8,  Are  hundreds  ? 
39.  Are  hundredths  ? 
J^O,  Are  millions  ? 
Ji-l.  Are  thousandths  ? 
4^.  Are  thousands  ? 
^<?.  Are  ten-millionths  ? 


^.  Are  ten-thousandths  ? 

Jf-5.  Are  millionths  ? 

^^.  Are  ten-thousands  ? 

^7.  Are  hundred-thousandths  ? 

^<^.  Are  ten-millions? 

Jf-Q.  Are  hundred-millionths  ? 


114,  When  the  right-hand  figure  of  a  decimal  ex- 
presses ten-thousandths,  the  whole  decimal  is  read  as  ten- 
thousandths  ;  when  the  right-hand  figure  expresses  hun- 
dred-thousandths, the  whole  decimal  is  read  as  hundred- 
thousandths  ;  and  so  on. 

Ten-thousandths  are  written       Hundred-thousandths  are  written 
.0001         .0005         .0008        .00001         .00004         .00007 
.0003  .0006  .0009         .00002  .00006         .00009 

115.  Principles  of  Decimal  Notation. 

I.  Ten  units  of  any  order  are  one  unit  of  the  next 
higher  order, 

II.  One  unit  of  any  order  is  ten  units  of  the  next 
lower  order,     (See  25.) 


A.   Copy  and  read 

B.  Write 

in  words 

.0002 

.0480 

.40003 

.9018 

80.96710 

.0056 

1.0006 

8.02904 

508.72023 

3,000.0845 

.1301 

20.0809 

9.06041 

7.00068 

101.03478 

.0400 

.00024 

34.21008 

.18000 

45.75692 

C.  Copy  and  read 

8.528040 

25.58 

0016 

.1413948 

59.00654387 

.4366 

576 

.25 

6153 

709. 

100365 

3, 

054.26405746 

X>.  Express  by  figures 

1,  5  ten  -  thousandths  ;  62  ten  -  thousandths  ;  207  ten- 
thousandths  ;  13  and  59  ten-thousandths. 

^.  Five  hundred  seventeen  and  three  thousand  six  hun- 
dred forty-seven  ten-thousandths. 


116  SECOND    BOOK  IN  ARITHMETIC. 

3,  Five  thousand  eighteen  hundred-thousandths. 

^.  Two  thousand  sixteen  ten-thousandths. 

5.  One  hundred  fifteen  ten-thousandths. 

6.  97  thousand  and  18  ten  -  thousandths ;  12  hundred- 
thousandths  ;  600  thousand  and  284  hundred-thousandths. 

7.  Six  hundred  thousand  two  hundred  four  millionths. 

8.  One  hundred  ninety-seven  ten-millionths. 

9.  Five  hundred  ninety-one  millionths. 

10,  Two  hundred  seven  thousand  fifty-six  ten-millionths. 

11,  46  million  274  thousand  508  hundred-millionths. 

12,  17  and  1,425  hundred-millionths. 

13,  Ninety  million  and  forty-one  ten-millionths. 

H,  Thirty-four  million  seventeen  hundred-millionths. 

no.  Rule  for  Decimal  Notation. 
Write  the  decimal  point,  and  after  it  the  figures  that 
express  the  units  of  each  order, 

117.  Rule  foe  Decimal  Numeration. 
Eead  the  decimal  as  an  integer^  and  give  to  it  the  name 
of  its  lowest  order. 

Exercises  in  Decimal  Notation  and  Numeration. 
Write  and  read  the  numbers  expressed  by 

1,  9  thousandths ;  23  thousandths ;  35  and  7  tenths ; 
4  and  276  thousandths. 

2,  9  and  18  ten -thousandths ;  55  and  5  hundredths; 
2,987  ten-thousandths ;  4  and  68,001  ten-millionths. 

3,  400,000  and  4  hundred-thousandths. 

^.  1  thousand  and  20  thousand  84  hundred-thousandths ; 
7  thousand  17  and  4  millionths. 

5,  78  thousand  and  219  millionths ;  106,204  hundred- 
millionths  ;  6  and  49  hundred-millionths. 


DECIMALS.^NOTATION  AND  NUMERATION.    117 

Express  by  figures 

6.  Forty -one  tliousand  thirty -six  and  two  hundred 
seventy-three  thousandths. 

7.  Fourteen  hundred-thousandths. 

8.  Forty-four  and  forty-four  hundred-thousandths. 

9.  Two  hundred  six  thousand  four  hundred  seventy 
and  three  tenths. 

10,  50  million  and  50  thousand  5  hundred  12  hundred- 
thousandths. 

11,  101  million  101  thousand  101  and  1  million  1  thou- 
sand 1  hundred-millionths. 

1^,  Sixty-one  million  thirty-two  thousand  eight  hun- 
dred seventy -six  and  two  million  eight  hundred  fifty 
thousand  forty-one  ten-millionths. 

13,  One  hundred  seventy -two  and  eight  tenths;  six 
hundred  twenty-four  hundred-thousandths. 

H„  One  hundred  fifty-two  million  six  hundred  fifty  and 
fifty  million  forty  thousand  thirty-six  hundred-millionths. 

15.  Sixty-five  thousand  and  seven  ten-thousandths. 

16,  Thirty  and  six  thousand  eight  ten-thousandths. 

118.  Currency  is  the  money  used  in  trade  and 
commerce. 

The  currency  of  the  United  States  is  a  decimal  cur- 
rency.    It  is  sometimes  called  Federal  Money, 

119,  The  money  units  in  common  use  are  the 
dollar^  the  ce7it.,  and  the  mill, 

a.  A  cent  is  1  hundredth  of  a  dollar  ;  and 

A  mill  is  1  tenth  of  a  cent,  or  1  thousandth  of  a  dollar.    Hence, 

10   mills   are  1  cent. 

100  cents   "    1  dollar. 

Write  1  mill,  $.001.  5  dollars  5Q  cents  8  mills,  $5,568. 

"      6  mills,  $  .006.  20  dollars  20  cents  1  mill,  $20,201. 

"      10  cents  5  mills,  $.105.    1  dollars  4  cents  3  mills,  $7,043. 


118 


SECOND   BOOK  IN  ARITHMETIC, 


%  .0006  is  6  tenths  of  a  mill ;  $  .0085  is  8  and  5  tentlis  mills. 

$  .2943  is  29  cents  4  and  3  tenths  mills. 

$15.65425  is  15  dollars  65  cents  4  and  25  hundredths  mills. 

b.  Write  cents  as  hundredths^  and  mills  as  thousandths^  of  a 

dollar. 

c.  Mead  decimal  parts  of  a  dollar  less  than  mills  as  decimals 

of  a  mill. 

1^0.  Halves,  fourths,  and  eighths  of  a  cent  are  exten- 
sively used  in  business  computations. 

i  cent  or  $.00 J  is  5   mills  or  $.005 


i 

'    -  l.ooj  - 

2.5    "     "  $.0025 

'     "   I.OOf  "  7.5    *'     **   $.0075 

'     ''  $.00J-  **  1.25  **      "   $.00125 

'     **   $.00f  *'  3.75  "      "   $.00375 

'     **   $.00f  ''  6.25  "      "   $.00625 

'     "  $.00^  '*  8.75  *'      "   $.00875 

Exercises. 

Copy  and  read 

Write 

1.  $.225 

I 

$300,567 

7.  8  mills;  15  cents  6  mills. 

^.  $.007 

5. 

$12,108 

8.  83  dollars  12  cents  5  mills. 

S.  $5,151 

6. 

$90,025 

9.  400  dollars  8  cents  1  mill. 

Write  in  decimal  form  and  read 

10.  12J  cents;  6^  cents;  18f  cents. 

11.  65f  cents ;  85f  cents ;  2|  cents ;  -J  cent. 

Write  the  decimal  parts  less  than  cents  in  fractional  form, 
and  read 

12.  %  .375 ;  $  .2025 ;  $1.09125 ;  $10.00625  ;  $105.56875. 
IS.  Seventy-five  dollars  thirty-seven  and  1  half  cents. 
H.  One  thousand  dollars  one  cent  one  mill. 

15.  Nine  dollars  nine  and  one  eighth  cents. 

16.  Fifteen  dollars  forty -three    cents   seven   and  five 
tenths  mills ;  four  dollars  three  and  one  eighth  cents. 

17.  Twelve  and  one  half  cents;  thirty -one  aijd  one 
fourth  cents. 


SECTIO]^   II. 

REDUCTION. 

ISl.  deduction  is  the  process   of  changing  num- 

l:4ers  from  given  to  required  units,  without  changing  their 

values. 

Changing  tenths  to  hundredths  or  thousandths ;  thousandths  to 
hundredths,  tenths,  or  ones ;  dollars  to  cents  or  mills ;  mills  or 
cents  to  dollars,  are  examples  of  reduction. 

The  same  values  are  expressed 

i.  By  .5,  .50,  and  .500.  .^.  By  .6300,  .630,  and  .63. 

2.  By  2,  2.0,  and  2.00.  5.  By  9.00,  9.0,  and  9. 

S.  By  7.3,  7.30,  and  7.300.         6.  By  3.400,  3.40,  and  3.4. 

1.33.  Keduction  of  decimals  is  based  upon  the  following 

Pkinciple.  Annexing  decimal  ciphers  to  a  numher^  or 
removing  decimal  ciphers  from  the  right  of  a  number^ 
does  not  change  its  value, 

Reduce 

i.  .7  to  hundredths. 
^.  .025  to  millionths. 

3.  62  to  ten-thousandths. 

4.  .930  to  hundredths. 

How  are  hundredths  j    9.  To  thousandths  ? 
reduced  ( 10.  To  ten-millionths  ? 


5.  .003000  to  thousandths. 

6.  800  hundredths  to  ones. 

7.  75  tenths  to  ones. 

8.  368  hundredths  to  ones. 

11.  To  tenths? 

12.  To  ones? 


13.  How  are  tenths,  hundred-thousandths,  thousandths, 
ones,  and  ten-thousandths  reduced  to  millionths  ? 


Reduce 

IJf.  5.3  to  thousandths. 

15.  .07  to  millionths. 

16.  18.602  to  ten-thousandths. 


Reduce  to  the  same  decimal  unit 

17.  1.2,  4.37,  192,  and  .0004. 

18.  30.251,  .0089,  and  3.000004. 

19.  6.0108,  57.8,  234,  and  2.34. 


120 


SECOND   BOOK  IN   ARITHMETIC, 


1^3,  The  same  values  are  expressed 


Jf.  By  200  cents  and  $2. 

5.  By  675  cents  and  $6.75. 

6,  By  953  mills  and  95.3  cents. 


1.  By  $5,  $5.00,  and  500  cents. 

2.  By  $5,  $5,000,  and  5,000  mills. 
8,  By  $  .37,  $  .370,  and  370  mills. 

7.  By  4,621  mills,  462.1  cents,  and  $4,621. 

8.  By  15  mills,  1.5  cents,  and  $  .015. 

9.  By  $31.25  and  3,125  cents.     Hence, 

In  decimal  currency, 

a.  To  reduce  dollars  to  cents: — Annex  two  ciphers, 

b.  To  reduce  dollars  to  mills : — Annex  three  ciphers, 

c.  To  reduce  cents  to  mills  : — Annex  one  cipher. 

d.  To  reduce  dollars  and  cents  to  cents ;  or  dollars,  cents, 
and  mills  to  mills: — Remove  the  dollar  mark  and  the  deci- 
mal point, 

e.  To  reduce  dollars  and  cents  to  mills: — Annex  one  cipher , 
and  remove  the  decimal  point. 

f.  To  reduce  cents  to  dollars : — Point  off  two  decimal  places^ 
and  prefix  the  dollar  mark. 

g.  To  reduce  mills  to  cents : — Point  off  one  decimal  place. 
h.  To  reduce  mills  to  dollars : — Point  off  three  decimal  places, 

and  prefix  the  dollar  m.ark. 

Reduce 


1.  93  dollars  to  cents. 

8.  57  cents  to  mills. 
5.  218  dollars  to  mills. 
7.  $  .40  to  mills. 

9.  $3  to  cents ;  to  mills. 
11.  $3.40  to  mills. 


Reduce  to  mills 
18.  $.09;  $.83. 
H.  $.213;  $.605. 
15,  $7;  $32;  $.082. 


2.  86,000  cents  to  dollars. 

^.  359  mills  to  cents. 

6.  9,274  mills  to  dollars. 

8.  6,000  cents  to  dollars. 
10.  3,219  mills  to  cents. 
12.  3,219  mills  to  dollars. 

Reduce  to  dollars 

19.  300  cents. 

20.  4,275  cents. 

21.  2,047  mills. 


Reduce  to  cents 
16.  $4;  $15. 
n.  $3.27;  $10.50. 
18.  875  mills. 


SECTION  III. 


ADDITION. 

1^4.  The  principles  for  addition  of  integers^  page  23, 
apply  equally  to  addition  of  decimals. 

Written  Work.-— En.  What  is  the 
sum  of  7.4875,  836.5,  .34,  85.075,  and 
973? 

Explanation. — I  write  the  numbers  so  that 

units  of  the  same  order  stand  in  the  same 

column. 
I  begin  at  the  right  and  add  as  in  integers, 

and  place  a  decimal  point  in  the  result 

under  the  decimal  points  in  the  parts. 
Always  place  the  decimal  point  in  the  result,  when  the  tenths  of  the 

result  are  written. 

Peoblems. 


Process. 

7. 

.lp875 

836. 

.5 

.3k 

85. 

.075 

973 

1,902. J^0%5 


.967 

.125 

tons 

800.1 

$121.10 

.00054 

1.25 

tons 

9.6Q 

38.47 

953.5 

12.5 

tons         2,064.25 

92.86 

7.375 

.0125 

tons 

167.4 

582.79 

6.75 

.00125  tons 

6                               7 

283 

810.04 

5 

8 

\     7.28 

$.58 

$2,000 

47.25 

days 

241.09 

.145 

5.75 

5.00695  days 

.42 

.0275 

48.01 

193.9 

days 

.96 

.5625 

.495 

5.876      days 

44.52 

.095 

359.17 

9.00005  days 

9.  A  silversmith  used  4.75  ounces  of  silver  in  making 
vases,  8.65  ounces  in  making  goblets,  and  10.6  ounces  in 
making  napkin  rings.     How  much  silver  did  he  use  ? 

F 


122  SECOND    BOOK  IN  ARITHMETIC. 

10.  A  family  used  .732  of  a  ton  of  coal  in  February, 
.824  of  a  ton  in  March,  .688  of  a  ton  in  April,  and  .595  of 
a  ton  in  May.     How  many  tons  of  coal  did  they  use  ? 

11.  In  four  meadows  containing  11.25  acres,  6.75  acres, 
13.33  acres,  and  8.Y2  acres,  are  how  many  acres  ? 

12.  A  boy  sold  3.75  bushels,  5.625  bushels,  9.5  bushels, 
and  6.25  bushels  of  pears.    How  many  bushels  did  he  sell? 

13.  A  seedsman  sold  to  five  farmers  2.25  bushels,  1  bush- 
el, 3.5  bushels,  4.75  bushels,  and  3.225  bushels  of  grass 
seed.     How  many  bushels  did  he  sell  ? 

IJf,.  Find  the  sum  of  two  hundred  and  four  hundredths, 
five  thousand  one  hundred  eight  and  seven  thousand  three 
millionths,  one  hundred  thirteen  thousand  seven  hundred 
six  and  two  thousand  five  hundred  four  ten-thousandths, 
and  ten  and  fifty-five  millionths. 

15.  By  selling  a  horse  for  $183.75,  I  lost  $24.50.  For 
how  much  should  I  have  sold  him,  to  gain  $39.70  ? 

16.  A  mechanic  earned  $56.25  in  January,  $45.63  in 
February,  $67.50  in  March,  $65.87^  in  April,  and  $75  in 
May.     How  much  did  he  earn  in  the  five  months  ? 

17.  A  lady  bought  3  dozen  buttons  for  $1.08,  2  yards 
of  ribbon  for  $  .37^,  6  yards  of  muslin  for  $1.18f ,  some 
needles  for  $  .31^,  a  belt  for  $  .75,  and  a  dress  for  $10.62|-. 
How  much  did  her  purchases  amount  to  ? 

18.  A  grocer  bought  six  hogsheads  of  molasses,  contain- 
ing 117.5  gallons,  124  gallons,  129.3175  gallons,  104.75 
gallons,  130.0625  gallons,  and  131.5625  gallons.  How 
many  gallons  of  molasses  did  he  buy  ? 

19.  What  is  the  sum  of  967  thousandths,  54  hundred- 
thousandths,  953  and  5  tenths,  7  and  375  thousandths, 
1,000  and  1  ten-thousandth,  6  and  75  hundredths,  8  and 
80,808  hundred-thousandths,  and  483  ? 


SECTION  lY. 

SUBTRACTION. 

1^5.  The  principle  for  subtraction  of  integers^  page 
40,  applies  equally  to  subtraction  of  decimals, 

Ex.  Subtract  16.78  from  38.25;    23.51  from  257.M2; 
and  93.1875  from  130.5. 

Processes. 
Explanation.  — I  ,  2  3 

write  the  numbers  _  _  _ 

in  each  example  so      S  8 .2  5      ^57AJp^      ISO, 5 
that  units  of  the      16.78        2  3.51  93.1875 

"^Z'u^LZ  ^^^  233.632  37.3125 
orders  in  the  minuend. 

I  Ibegin  at  the  right  and  subtract,  as  in  integers,  and  place  a  deci- 
mal point  in  the  result  under  the  decimal  point  in  the  minuend 
and  subtrahend. 

In  the  third  process,  I  suppose  three  decimal  ciphers  to  be  an- 
nexed to  the  minuend,  when  I  perform  the  subtraction. 

Always  place  the  decimal  point  in  the  result,  when  the  tenths  of  the 
result  are  written. 

Problems. 

1^34  5 

325.48  87.006  1.03045  374  20,000.2875 

199.56  9.84  .0009  .125  482.52006 


$250.35  $.104  $100,000  $1,000  $50 

187.50  .087  5.875  .065  2.875 


11.  From  a  piece  of  linen  containing  43.5  yards,  a  clerk 
sold  26.8  yards.     How  many  yards  remained? 

12.  362.413  miles- 81.0064  miles = how  many  miles? 


124  SECOND    BOOK  IN  ARITHMETIC. 

13.  At  night  the  snow  was  11.37  inches  deep,  and  in 
the  morning  it  was  13.03  inches  deep.  How  much  snow 
fell  during  the  night  ? 

IJf..  Of  a  railroad  240.475  miles  long,  168.025  miles  are 
double  track.     How  many  miles  are  single  track  ? 

15.  From  a  barrel  of  kerosene  containing  42  gallons, 
31.25  gallons  were  drawn.     How  many  gallons  remained? 

16.  Two  men  built  134  rods  of  stone  fence,  one  of  them 
building  65.87  rods.    How  many  rods  did  the  other  build? 

17.  From  a  5-pound  box  of  starch,  a  grocer  sold  2.0625 
pounds.     How  many  pounds  remained  in  the  box  ? 

18.  From  ninety-five  and  eight  thousand  five  ten-thou- 
sandths take  ten  and  forty-four  hundredths. 

19.  A  tree  79.95  feet  high  broke  off  4.7  feet  above  the 
ground.     What  was  the  length  of  the  part  broken  off  ? 

W.  A  silver  dollar  weighs  412.5  grains,  and  contains 
41.25  grains  of  copper.  How  much  pure  silver  does  it 
contain  ? 

21.  From  eight  hundred  sixty  and  four  hundredths  take 
nineteen  and  nine  thousand  fifty-four  hundred-thousandths. 

22.  A  cubic  foot  of  gold  weighs  1,203.625  pounds,  and 
a  cubic  foot  of  iron  450.4375  pounds.  How  much  more 
does  the  gold  weigh  than  the  iron  ? 

23.  If  wheat  is  worth  $1.18f  per  bushel  in  Milwaukee 
and  $1.87^  in  ]N"ew  York,  how  much  is  added  to  its  value 
by  transportation  ? 

2J,,.  In  a  cistern  that  will  hold  320.5  barrels,  are  192.8125 
barrels  of  water.  How  much  more  water  will  the  cistern 
hold? 

25.  Mr.  Brown  exchanged  a  silver  watch  worth  $18.75, 
for  a  gold  watch  worth  $80,  paying  the  difference  in 
money.     How  much  money  did  he  pay '^ 


DECIMALS.— SUBTRACTION.  125 

^6.  A  merchant  sold  a  piece  of  damaged  cloth  that  cost 
him  $94.37,  at  a  loss  of  $26.75.   For  how  much  did  he  sell  it  ? 

^7.  A  vessel  sailed  from  Portland,  Me.,  for  Mobile,  with 
a  cargo  of  1,438.275  tons  of  ice,  and  561.895  tons  of  it 
melted  on  the  voyage.  How  much  more  ice  reached  Mo- 
bile than  melted  on  the  voyage  ? 

^8,  A  merchant  bought  a  jar  of  butter  for  $15.37|-,  pay- 
ing $6.87^  in  cloth,  $4.62|-  in  groceries,  and  the  balance  in 
money.     How  much  money  did  he  pay  ? 

^9.  A  farmer  raised  32.25  bushels  of  timothy-grass  seed. 
He  sowed  4.03125  bushels,  and  sold  12.125  bushels.  How 
many  bushels  had  he  left  ? 

30.  A  laborer  received  $6  for  his  week's  work.  He  paid 
$1.62-1-  for  flour,  $  .85  for  tea,  $  .75  for  sugar,  and  $  .37|  for 
butter.     How  much  money  had  he  left  ? 

31.  From  an  ice-house  containing  500  tons  of  ice,  a  deal- 
er sold  18.263  tons,  15.967  tons,  17.4  tons,  and  16.48  tons. 
How  much  ice  was  left  in  the  ice-house  ? 

32.  From  a  cask  containing  37.175  gallons  of  alcohol, 
a  druggist  drew  at  different  times  .125  of  a  gallon,  1.5 
gallons,  .25  of  a  gallon,  and  .75  of  a  gallon.  How  many 
gallons  remained  in  the  cask  ? 

33.  A  butcher  killed  an  ox  that  cost  him  $63.37.  He 
retailed  the  meat  for  $61.96,  sold  the  tallow  for  $6.08,  and 
the  hide  for  $8.55.     How  much  were  his  profits  ? 

3Jf.  C  has  53.843  acres  of  land  in  one  field,  75.364  acres 
in  another,  and  15.527  acres  in  a  third.  How  much  must 
he  purchase  of  a  neighbor,  to  have  a  farm  of  200  acres  ? 

35.  B  invests  $1,750.25  in  oats,  $786.37-1-  in  corn,  and 
$2,648.62^  in  wheat.  He  sells  the  oats  for  $2,022.45,  the 
corn  for  $831.50,  and  the  wheat  for  $2,331.30.  Does  he 
gain  or  lose,  and  how  much,  on  the  oats  ?  On  the  corn  ? 
On  the  wheat  ?     On  the  three  investments  ? 


SECTION  V. 

MULTIPLICATION. 

ISO.  The  methods  of  written  work  in  multiplication 
of  decimals  are  based  upon  the  following 

Principle.  There  must  he  as  many  decimal  places  in 
the  product  as  there  are  in  hoth  factors, 

Ex.  1.  The  factors  are  .24  and  39.   What  is  the  product? 

Explanation. — I  write  the  First  Process.    Second  Process. 

factors  and  multiply  as  in  o  i  o  g 

integers ;   and,  since  one  qq  6>  i 

factor  is  an  integer,  and  ^ *^  '^ ^ 

there    are    two    decimal  ^16  16  6 

places  in  the  other  factor,  ry  q  ^  n 

I  point  off  two  decimal  *        ^  

figures  in  the  product.  9.36  9.3  6 

When  one  factor  is  an  integer,  the  product  has  the  same  number  of 
decimal  places  as  the  otlier  factor, 

Ex.  2.  Multiply  5.63  by  .8.  Pj^ocess. 

Explanation.— I  write  the  numbers  and  multiply  S  63 

as  in  integers;  and,  since  there  are  three  decimal  *     ^ 

places  in  the  factors,  I  point  off  three  decimal  fig-       l£ 

ures  in  the  product.  ^..S 0^ 
Problems. 

12  3  4  5 


210.735 

634.04 

.19125 

250,375 

47,232 

9 

76 

108 

.07 

7.38 

6 

7 

8 

9 

10 

$194.17 

$310.75 

$5,044 

$24,960 

$74,809 

18 

n 

236 

7.5 

.25 

17.05 

12 

13 

14 

15 

Multiply 

4.29 

9432.6 

$647.89 

$296.07 

5.01234 

by 

.27 

55.55 

20.009 

65.33 

.8007 

DECIMALS.—MUL  TIPLICA  TION.  127 

16.  How  many  feet  is  it  across  a  street  5  rods  wide  ? 

(1  rod  is  16.5  feet.) 

17.  How  many  bushels  of  oats  are  there  in  4  bins,  each 
bin  containing  20.75  bnshels  ? 

18.  How  many  days  are  there  in  10.7  years  ? 

19.  What  is  the  weight  of  47  reams  of  printing-paper, 
each  ream  weighing  38.125  pounds  ? 

^20.  Of  37  thousand  feet  of  lumber,  @  28.25? 
xy-i    X  •       *^i.  Of  29  rolls  of  wall  paper,  @  44c.? 

the  value  1  ^^'  ^^  ^^^  P^^'^^'  ""^  ^^^^''  ®  ^^- ' 

^^.  Of  148  yards  of  broadcloth,  @  $3.87|-? 

^21^.  Of  55.3  acres  of  land,  @  $118  ? 

25.  If  one  ton  of  lead  ore  yields  .765  of  a  ton  of  lead, 
372.084  tons  of  ore  will  yield  how  many  tons  of  lead  ? 

26.  How  many  tons  of  flax  can  be  raised  on  .85  of  an 
acre  of  land,  if  1.8764  tons  are  raised  on  one  acre  ? 

27.  If  .35  of  a  pound  of  butter  is  made  from  1  gallon 
of  milk,  how  many  pounds  of  butter  can  be  made  from 
2,245.5  gallons  of  milk? 

28.  If  22.73  gallons  of  brine  are  required  for  1  bushel 
of  salt,  how  many  gallons  are  required  for  83.25  bushels  ? 

1^7.  Ex.  Multiply  .75  by  .003. 

Process 
Explanation. — I  write  the  numbers  and  multi- 
ply as  in  integers ;  and,  since  there  are  five  deci-  .  7  5 
mal  places  in  the  factors,  I  prefix  two  decimal                  0  0'^ 

ciphers  to  the  product  of  75  and  3,  thus  making         '- 

the  number  of  decimal  places  in  the  product  .00226 

equal  to  the  number  in  both  factors. 
When  there  are  not  as  many  figures  in  the  written  result  as 
there  are  decimal  places  in  the  factors^  supply  the  deficiency  by 
prefixing  decimal  ciphers. 

Problems. 

J[  2  3  4  5 

.0854       .084       .00393       .00049       .06052 
.032        .07        .006        .057        .066 


128  SECOND    BOOK  IN  ARITHMETIC. 

6.  How  much  grass  seed  will  be  required  for  .05  of  an 
acre,  at  the  rate  of  .3  of  a  bushel  to  the  acre  ? 

7.  How  far  will  a  locomotive  run  in  .25  of  a  minute,  at 
the  rate  of  .36  of  a  mile  per  minute  ? 

8.  How  much  will  .375  of  a  pound  of  bicarbonate  of 
soda  cost,  at  $  .18|  a  pound  ? 

9.  Find  the  cost  of  15.5  yards  of  tape,  at  $  .OOJ  a  yard. 

10,  A  cook  used  .09  of  .31  of  a  gallon  of  molasses.   How 
much  molasses  did  she  use  ? 

11.  Multiply  32  ten-thousandths  by  7  hundredths. 

138.  Ex.  Multiply  35.827  by  10,  by  100,  by  1,000,  and 

by  100,000. 

Processes. 

35,827x  10=  358.270=  358.27 

35.827X  100=  3,582.700=         3,582.7 

35.827X       1,000=       35,827.000=       35,827 
35.827x100,000  =  3,582,700.000=3,582,700 

Each  removal  of  *  the  decimal  point  in  a  number  one  place  to 
the  rights  multiplies  the  number  by  10.     (See  S5.) 

Problems. 
Multiply 


5.  $  .0625  by   100. 

6.  $37.81i  by  10,000. 


1.  53.78  by  10.  3.     6.8794  by  100. 

2.  $2.31  by  10.  Jf.   59.6043  by  1,000. 

7.  What  is  the  total  length  of  1,000  bars  of  railroad 
iron,  each  bar  being  .00062  of  a  mile  long? 

i    8.  Of  1,000  fat  cattle,  @  $42.69}. 
Find  the  cost  I    9.  Of  100  days'  board,  @  75c. 

(  10.  Of  10,000  quart  bottles,  @  7^0. 

139.  Rule  for  Multiplication  of  Decevials. 
I.   Write  the  numhers  and  Tnultiply  as  in  integers. 
II.  Point  off  as  many  decimal  figures  m  the  jp7vduct 
as  there  are  decimal  jplaces  in  hoth  factors. 


DECIMAL8.---MULTIPLICA  TION.  129 

Problems. 
i.  Multiply  172.84  by  50. 


2.  Multiply  $2.73  by  8,600, 
^.Multiply  .12815  by  93.7. 

4.  Multiply  $.84|-by  .946. 

5.  Multiply  .0062  by  .0008. 


6.  400,000  X. 00004 =how  many? 

7.  397,654  X  380.07 =how  many? 

8.  $100,000  x5.875  =  how  many? 

9.  2,500  X. 25  rrr  how  many? 
10,  .000007  x7,000,000  =  how  many? 

11.  How  much  silk  will  be  required  for  9  yards  of  rib- 
bon, if  .085  of  a  pound  is  required  for  one  yard  ? 

12.  How  many  yards  are  there  in  25  pieces  of  tapestry 
carpeting,  each  piece  containing  62.75  yards  ? 

IS.  How  many  rods  of  stone  fence  will  a  man  lay  in 
128  days,  if  he  lays  3.69  rods  each  day? 

i^.  If  a  rolling-mill  makes  94.2  tons  of  iron  per  day, 
how  many  tons  will  it  make  in  164.25  days  ? 

15.  If  3.75  gallons  of  cider  can  be  made  from  one  bushel 
of  apples,  how  much  cider  can  be  made  from  38.5  bushels  ? 

Find  the  cost 

16.  Of  .84  of  a  ton  of  plaster,  at  $4.25  a  ton. 

17.  Of  15,000  bushels  of  wheat,  at  $1.06|  a  bushel. 

18.  Of  18.4  tons  of  straw,  at  $3.12^  a  ton. 

19.  Of  7  pieces  of  lace,  39  yards  each,  at  $  .37|-  a  yard. 

20.  How  much  is  the  freight  on  .456  of  a  ton  of  goods 
from  New  York  to  Toledo,  by  railroad,  at  $28.60  a  ton  ? 

21.  What  is  the  value  of  a  million  pins,  at  one-ten-thou- 
sandth of  a  dollar  each  % 

22.  I  bought  2.5  yards  of  broadcloth,  at  $3.75  f  1  yard 
of  cashmere  for  $  .87|- ;  26  yards  of  calico,  at  $  .12|- ;  and  14 
yards  of  muslin,  at  $  .35.    What  was  the  cost  of  the  whole  ? 

2S.  A  mechanic  earns  $2.75  per  day,  and  his  expenses  . 
are  $1.40  per  day.     How  much  does  he  save  in  a  week  ? 

21^..  I  bought  42.5  gallons  of  linseed-oil  for  $37.18f,  and 
sold  it  at  $1.12|-  per  gallon.     How  much  did  I  gain  ? 

r2 


SECTION  VI. 

DIVISION. 

130.  The  methods  of  written  work  in  division  of  deci- 
mals  are  based  upon  the  following 

Peinciple.  There  must  he  ds  many  decimal  places  in 
the  quotient  as  the  numher  of  decimal  places  in  the  divi- 
dend exceeds  the  number  in  the  dvvisor, 

Ex.  Divide  1,947.15  by  9.  ^^^^ 

Explanation.— I  divide  as  in  integers;  and,  n  i  rf 

since  there  are  two  decimal  places  in  the  )  ^     "^  7,lo 

dividend  and  none  in  the  divisor,  I  point  2 16,35 
off  two  decimal  places  in  the  quotient. 

When  the  dimsor  is  an  integer,  the  quotient  has  the  same  number  of 
decimal  places  as  the  dividend. 


Problems. 

2                                   3 

4 

39).897(             55)33.440( 

105)7.89285( 

9). 396 

5.  My  horse  eats  .324  of  a  ton  of  hay  in  9  weeks.    How 
much  hay  does  he  eat  in  a  week  ? 

6.  If  a  boarding -house  keeper  uses  2.1875  barrels  of 
flour  in  a  week,  how  much  flour  does  she  use  in  a  day  ? 

7.  A  fruit  dealer  sold  686  boxes  of  oranges  for  $1,543.50. 
What  was  the  price  per  box  ? 

8.  A  gentleman  bought  a  parlor  carpet  containing  43 
yards,  for  $96.75.     How  much  did  he  pay  per  yard  ? 

9.  If  55  peach  baskets  hold  34.375  bushels,  how  much 
does  each  basket  hold  ? 


DECIMALS.— DIVISION.  131 

131.  Ex.  Divide  15.695  by  7.3.  Peocess. 

Explanation.— I  divide  as  in  integers ;  7.3)15.6  9  5 (^2.15 

and,  since   there   are    three   decimal  1^6 

places  in  the  dividend  and  one  deci-  10  9 

mal  place  in  the  divisor,  I  point  off  ^  ^ 

two  decimal  places  in  the  quotient,  '  "^ 

and  I  have  2.15,  the  required  quo-  36  5 

tient.  ^  ^  5 


Note. — To  secure  accuracy  in  pointing  off  decimal  figures  in  the  quo- 
tient, many  teachers  prefer  the  following  method  : 

1.  Before  beginning  to  divide,  cut  off,  by  a  line,  as  many  7.3)  15.6\95{2.15 
places  to  the  right  of  the  decimal  point  in  the  dividend  as  1^^ 

there  are  decimal  places  in  the  divisor,  as  shown  in  the  ex-  10  9 

ample  in  the  margin.  7  3 

2.  In  dividing,  write  the  decimal  point  in  the  result  wfien  365 
all  the  orders  to  the  left  of  the  line  have  been  v^ed.  ^^^ 

Peoblems. 
1,  How  many  times  is  16.54  contained  in  611.98  ? 
^.  What  is  tlie  quotient  of  .3264  divided  by  .0034  ? 

3.  If  one  barrel  of  flonr  is  made  from  4.5  bushels  of 
wheat,  how  many  barrels  will  be  made  from  67.5  bushels  ? 

4^.  How  many  sheets  of  Russia  iron  weighing  3.31  pounds 
each,  will  a  workman  use  in  making  76.13  pounds  of  stove- 
pipe? 

5.  If  one  overcoat  is  made  from  3.25  yards  of  cloth,  how 
many  overcoats  can  be  made  from  61.75  yards  ? 

6.  How  many  casks,  each  holding  41.315  gallons,  will 
be  required,  to  hold  11,278.995  gallons  of  alcohol? 

7.  15.9392  tons  of  iron  ore  will  make  how  many  loads, 
each  weighing  .9376  of  a  ton? 

8.  A  carriage  ironer  paid  $24.72^  for  bar  st^el,  at  $  .07| 
a  pound.     How  many  pounds  of  steel  did  he  buy  ? 

9.  If  an  acre  of  land  produces  1.92  tons  of  hay,  how 
much  land  will  produce  .9024  of  a  ton  ? 


132  SECOND   BOOK  IN  ARITHMETIC. 


Process. 


13^.  Ex.  1.  Divide  18.24  by  6.4. 

Explanation. — I  divide  as  in  integers; 
and  when  all  the  figures  expressing         ^-^  )  ^  ^'^  k  (  ^'O  ^ 

the  dividend  have  been  used,  I  form  12  8 

a  new  partial  dividend, — by  annex-  f:  f  f 

ing  a  cipher  to  the  remainder, — and  t>  4  4. 

continue  the  division.  612 

Counting  the  cipher  annexed  to  form  3  2  0 

the  last  partial  dividend  as  a  decimal  '^2  0 

place  of  the  given  dividend,  I  point  

off  two  decimal  places  in  the  quotient. 

The  result,  2.85,  is  the  required  quotient. 

a.  When  there  is  a  remainder  after  all  the  orders  of  the  dividend 
have  been  used,  form  a  new  partial  dividend  by  annexing  a  cipher^ 
and  continue  the  division. 

b.  Decimal  ciphers  annexed  to  form  a  partial  dividend,  must  be 
counted  as  decimal  plaices  of  the  given  dividend. 

6.U)18.2\U{S.85 
Note.— Proceeding  as  directed  in  Note,  page  131,  the  l^S 

decimal  point  is  placed  in  the  quotient  before  writing  the  5UU 

first  decimal  figure.  SIS 

sso 
Problems.  s^o 


7.  .7854-4-. 56     = 

8.  .7854-r-.056  = 

9.  .7854-r-.0056  = 


X    7854^5.6=  4'  7.854-5.6  = 

2.  785.4-^5.6zz:  5.  .7854-f-5.6  = 

S.  78.54^5.6=  6:  .7854-7-56  = 

10.  Divide  76.94  by  4.9. 

76.94.  -^  19  =  15. 702 j^^  ;  or  76. 9 J,.  -^  ^.9  =  15.702^. 

When  the  division  does  not  terminate,  or  wh£n  it  has  been  carried  a» 
far  as  is  desirable,  express  the  remainder  fractionally,  as  a  part 
of  ths  quotient;  or,  write  +  after  the  quotient. 

Divide,  carrying  the  division  to  three  decimal  places, 
11.   1.264  by  3.   |  12.  3,156.293  by  25.17.   |  13.  $650  by  313. 

U.  A  baker  paid  $103.60  for  21.25  cords  of  wood. 
How  much  did  he  pay  per  cord? 

15.  A  manufacturer  made  25  sets  of  table-spoons  that 
weighed  231  ounces.     How  much  did  one  set  weigh  ? 


DECIMALS.— DIVISION.  133 

16.  If  .38  of  a  bushel  of  grass  seed  will  seed  one  acre  of 
land,  15.39  bushels  will  seed  how  many  acres  ? 

17.  An  Iowa  farmer  raised  3,045  bushels  of  corn  from 
56  acres.     How  much  was  the  yield  to  the  acre  ? 

18.  A  bookseller  sold  25  arithmetics  for  $8.43.     How 
much  did  he  receive  apiece  for  them  ? 

19.  Divide  an  estate  of  $31,723.25  equally  among  7 

heirs. 

Process. 

133.  Ex.  Divide  42  by  .56.  .66)4.2.00(75 

Explanation. — Since  there  are  two  deci-  3  9  2 

mal  places  in  the  divisor  and  none  in  2  8  0 

the  dividend,  I  annex  two  decimal  ci-  6)  q  n 

phers  to  the  dividend  before  dividing.  '^  "  ^ 

There  must  be  at  least  as  many  decimal  places  in  tJhe  dividend  as  in 
the  divisor,  before  beginning  to  divide. 

Problems. 

1.  Divide  5,463.9  by  42.02. 

2.  $875  -f-  $3.12|-  =  how  many  ? 

3.  A  farmer  exchanged  15  bushels  of  wheat  for  flour, 
receiving  1  sack  of  flour  for  every  1.25  bushels  of  wheat. 
How  many  sacks  of  flour  did  he  receive  % 

Jf,.  If  a  factory  girl  weaves  6.25  yards  of  sheetings  in 
an  hour,  in  how  many  hours  can  she  weave  40  yards  ? 

6.  When  the  price  of  rice  is  $  .06|-  a  pound,  how  many 
pounds  can  be  bought  for  $3.50  ? 

6.  A  stationer  sold  .875  of  a  gallon  of  ink,  in  bottles 
holding  .0625  of  a  gallon  each.  How  many  bottles  of  ink 
did  he  sell? 

7.  A  druggist  puts  up  23  gallons  of  cologne  in  736 
bottles.     How  much  cologne  does  each  bottle  contain  ? 


134:  SECOND   BOOK  IN  ARITHMETIC. 

134.  Ex.  Divide  .002  by  .08.  Process. 
Explanation.  —  After  dividing,  I  find  that      ,08). 0 0^ 

there  are  five  decimal  places  in  the  dividend  ^ iT^T^ 

used,  and  two  in  the  divisor.     T  therefore  .0  Jo 

point  off  three  decimal  places  in  the  result. 

Note. — The  process  is  more  readily  performed  by  the  .08).00\s 
method  given  in  Note, page  131.  .OS 5 

Problems. 

1.  Divide  .897  by  39.       2.  .27  by  4.32.       S.  .7  by  112. 

.4,  .0016  by  .612.    5.  .07  by  12.8.     ^.  2  by  53.1.  ^7.  .1537  by  29. 

8,  Divide  $8  into  200  equal  parts. 

9.  If  .Y505  of  an  ounce  of  gold-leaf  will  cover  79 
square  f eet^  how  much  gold-leaf  will  gild  one  square  foot  ? 

10.  $2,471^  for  275  lemons  is  how  much  apiece  ? 

135.  Ex.  1.  Divide  582.7  by  10,  by  100,  by  1,000, 
and  by  100,000. 


58^.7- 
682.7- 
682.7 
682.7 


Processes. 

10z=r.68.27 

100=    6.827 

1,000=       .6827 

100,000=       .006821 


Each  removal  of  a  decimal  point  in  a  number  otic  place  to  the  left 
divides  the  number  by  10. 

Ex.  2.  Divide  3,725.4  by  700. 
Explanation. — Since  700=7  times  100, 1  Process. 

divide  the  dividend  by  100,  and  obtain  ^y  f) n\  '^^\(p  Pi    J 

37.254;  and  this  result  I  divide  by  7,  and  ^   ^^  )0_r\^jD^ 
obtain  5.322,  the  required  result.  6.822 

Problems. 

1.  Divide  537.8  by  10.  Jf..  $.75  —  100  = 

2.  Divide  62.5  by  50.  5.  $437.50^1,000  = 

3.  Divide  5,960.43  by  900.  6.  $30,943.20-^$15,000  = 
7.  434.5  acres  of  land  were  divided  equally  among  10 

persons.     How  much  land  had  each  person  ? 


DECIMALS.—DIVISION,  135 

8.  An  Ohio  fanner  raised  2,175  bushels  of  corn  from 
40  acres  of  land.     How  mnch  was  the  yield  to  the  acre  ? 

9,  In  how  many  months  will  a  man  whose  wages  are 
$100  per  month,  earn  $937.50  ? 

10.  A  railroad  200  miles  long  was  constructed,  at  a  cost 
of  $8,263,860.50.     What  was  the  cost  per  mile  ? 

136.  Rule  foe  Division  of  Decimals. 
I.  Annex  decimal  ciphers  to  the  dividend^  when  neces- 
sary^ and  divide  as  in  integers, 

11.  Point  off  as  many  decimal  figures  in  the  quotient 
as  the  nuniber  of  decimal  places  in  the  dividend  exceeds 
the  number  in  the  divisor. 

a.  Count  decimal  ciphers  annexed  to  form  a  partial  dividend,  as  deci^ 
mat  places  of  the  given  dividend. 

b.  It  is  necessary  to  annex  decimal  ciphers  to  the  dividend 

1.  When  it  has  a  less  number  of  decimal  places  than  the  divisor. 

2.  When  the  dividend,  considered  as  an  integer,  is  less  than  the  dimsor 

considered  as  an  integer. 

3.  When  there  is  a  remainder  after  all  the  orders  of  the  dividend  have 

been  used. 

Problems. 
1.  .1632  divided  by  3.4=how  many  ? 
^.  How  many  times  is  .0753  contained  in  1,385.52  ? 
S.  What  is  the  quotient  of  5  divided  by  64  ? 
4-.  Divide  90.5  into  14,480  equal  parts. 

5.  639  cords -^  11. 25= how  many  cords? 

6.  15.39  bushels  are  how  many  times  .38  of  a  bushel  ? 

7.  How  many  pounds  of  wool  will  be  required  for  264 
yards  of  cloth,  at  the  rate  of  1  pound  for  .625  of  a  yard? 

8.  If  in  an  hour  18.75  barrels  of  water  run  into  a  cis- 
tern that  holds  204  barrels,  in  how  many  hours  will  the 
cistern  be  filled  ? 


136  SECOND   BOOK  IN  ARITHMETIC, 

9,  In  what  time  will  a  railroad  train  run  16  miles,  if  it 
runs  at  the  rate  of  31.25  miles  an  hour? 

10.  I  raised  147  bushels  of  oats  on  2.24  acres  of  land. 
At  what  rate  did  the  land  yield  per  acre  ? 

11.  If  one  gallon  of  sap  makes  .18  of  a  pound  of  maple 
sugar,  how  many  gallons  of  sap  will  make  112.59  pounds 
of  sugar  ? 

12.  If  1  bushel  of  charcoal  is  made  from  .0196  of  a 
cord  of  wood,  how  many  bushels  can  be  made  from  5.831 
cords  ? 

13.  $564.37^  for  105  sheep  is  how  much  a  head  ? 
H.  $35  buys  how  many  pounds  of  sugar,  at  $.12J? 
16.  $172.50  buys  how  many  lounges,  at  $7.50? 

16.  $59.06|-  for  31.5  gallons  is  how  much  per  gallon? 

17.  $1.87^  buys  how  many  bushels  of  plums,  at  $2.50  a 
bushel  ? 

18.  At  $  .56  a  yard,  $24.36  buys  how  many  yards  of 
muslin  ? 

19.  How  many  goblets,  each  weighing  7.5  ounces,  can 
be  made  from  176  ounces  of  silver? 

W.  815,105  pounds  of  hay  are  how  many  tons  of  2,000 
pounds  each,  and  how  many  pounds  over  ? 

21.  124.2  tons  of  coal  will  make  how  many  full  car  loads 
of  9.5  tons  each? 

22.  134  bushels  of  wheat  will  pay  for  how  many  sheep, 
at  2.5  bushels  for  1  sheep? 

23.  I  sold  .95  of  an  acre  of  land  in  building-lots  of  .125 
of  an  acre  each.     How  many  lots  did  I  sell  ? 

2 If..  A  baker  has  834.25  pounds  of  lard,  and  he  uses 
23.75  pounds  daily^.  How  many  full  days  will  the  lard 
last  him,  and  how  many  pounds  over  ? 


DECIMALS.— DIVISION.  137 

Accounts  and  Bills. 

137.  A  debt  is  money,  goods,  or  services  due  from 
one  party  to  another. 

a,  A  debtor  is  a  person  from  whom  a  debt  is  due. 

b,  A  creditor  is  a  person  to  whom  a  debt  is  due. 

138.  A  hill  of  goods  is  a  written  statement  given 
by  the  seller  to  the  buyer,  containing  the  date  of  the  pur- 
chase ;  the  names  of  buyer  and  seller ;  a  list  of  the  goods 
bought,  with  their  prices ;  and  the  total  amount  or  cost. 

a.  An  item  is  each  particular  in  a  bill. 

&•  Extending  an  item  is  finding  the  cost  of  the  item. 

c,  The  footing  is  the  total  cost  of  all  the  items  in  a  bill. 

139.  An  account  fin  business  transactions,  is  a  writ- 
ten  statement  of  debits  and  credits  between  two  parties. 

a.  The  balance  of  an  account  is  the  difference  between 

the  sums  of  the  debits  and  credits. 

b.  When  a  bill  or  account  is  paid,  the  party  to  whom  the 

payment  is  made  should  write  at  the  bottom  of  the  same, 
Received  payment,  or  Paid,  followed  by  his  name. 

Problems. 
Extend  the  items,  and  find  the  footings  in  the  following 
bills : 

1'  Buflalo,  ^€me  //,  /<^(^^. 

Mr.   QyZtlnut  QA^e^6^n^a7t 

SSouad^:  o^  B.  F.  Brown  &  Co. 

4  /lafiez^    ^atc/en    <Mec/d,  %  0    JO    .     .    fi 

5  cQa?nh  ^mmne'u<),  "  .Op    .    . 

/   ^oiTi  '^cc/uvaiot,  "       9.^5 

/    (^/lade.  /  .75/    /    ^/lac/i^^   ^otd,   //.J'J 

S    '^ti'oe^,  0  .63  anc/ ^  .^'f 


138  SECOND    BOOK  IN  ARITHMETIC. 

2,  CleTeland,  e^^f.   6,  ^^^S. 

Mr-   ^ame<}  "W^.    ^za^a??z/ 

^ou^^  o^  Henry  Arnold, 

5  ^  Jd^  "^^Z^'  @  J9^(f  .    .    .    ^ 
¥0  "     ^tac4eui,  "    //^    .    .    . 

6  "    ^taut,  ■  '*    so(p  .  .  . 

^  c/ox.    0iafi^a, .  J><^  , 

/  €0.  ^a^n  <^ea, 'f.'fO 

/   ^M//ot<.  ^a^4e^, S5 


S^ec  (/  ^ay^nerU, 


tty   Q^iTiotc/. 


Put  the  following  narrations  of  business  transactions 
into  bills  of  proper  forms  : 

S,  Philadelphia,  December  6,  1882,  Charles  Eoberts  & 
Co.  sold  to  Mrs.  Eliza  Williams,  for  cash,  2  doz.  silver  ta- 
ble forks,  at  $37.50  per  dozen ;  1  dozen  silver  table-spoons 
for  $33 ;  3  sets  of  silver  tea-spoons,  at  $9.25  per  set ;  and 
1  silver  cake  basket  for  $37.50. 

^.  Albany,  N.  T.,  JSTovember  1,  1882,  Joseph  Daniels 
bought  of  James  Eiley  &  Co.  15.5  tons  of  stove  coal,  at 
$4.75;  19  tons  of  grate  coal,  at  $5.12^;  and  14  cords  of 
wood,  at  $5.25. 

5,  Springfield,  December  24, 1881,  Jones  &  Bogart  sold 
to  Lyman  A.  Moore,  on  account,  1  piece  of  Lonsdale  cot- 
ton, 42  yards,  at  $  .12^ ;  5  yards  of  French  broadcloth,  at 
$4.75  ;  16  yards  of  Merrimac  prints,  at  $  .09 ;  8  yards  of 
Irish  linen,  at  $.68f ;  and  12  yards  of  Hamburg  edging, 
at  $  .37i. 

Note. — For  outlines  of  decimals  for  review,  see  page  271. 


DECIMALS.— REVIEW.  139 

General  Eeview  Problems  in  Decimals. 

1,  Multiply  twelve  thousandths  by  twelve  hundredths, 
and  divide  the  product  by  six  ten-thousandths. 

^.  From  nine  hundred  and  nine  ten-thousandths  sub- 
tract nine  hundred  nine  ten-thousandths,  and  divide  the 
remainder  by  nine  hundred  thousandths. 

3.  What  is  the  quotient  of  .01  divided  by  12.8? 

Jf,.  If  a  clerk  wishes  to  save  $100  a  year  out  of  a  salary 
of  $900,  how  much  can  he  spend  per  week  ? 

6.  If  105.6  tons  of  rails  are  required  for  1  mile  of  rail- 
road track, how  many  tons  are  required  for  137.55  miles? 

6.  How  much  is  the  freight  on  a  cargo  of  25,380  bush- 
els of  wheat,  from  Chicago  to  Buffalo,  at  $  .044  P^r  bushel? 

7.  A  half-dime  a  day  is  how  much  a  year  ? 

8.  31  laborers  received  $1,046.25  for  working  22.5 
days.     What  were  the  average  daily  wages  ? 

9.  When  tomatoes  are  worth  $3  a  bushel,  what  part  of 
a  bushel  can  be  bought  for  $  .3T-J  ? 

10.  I  sowed  2.736  quarts  of  clover  seed  in  my  orchard, 
at  the  rate  of  5.76  quarts  to  the  acre.  How  much  land 
is  there  in  my  orchard  ? 

11.  The  taxes  paid  in  a  certain  school-district  one  year 
were  as  follows:  By  A,  $38.75;  B,  $10.50;  C,  $132.50;  D, 
$6 ;  E,  $58.75 ;  F,  $2.50 ;  Glass  Manufactory,  $1,057.50 ;  H 
$30 ;  I,  $1.50 ;  J,  $8.60  ;  K,  $141.80 ;  L,  $3.75  ;  M,  $530 ' 
I^ational  Bank,  $1,515  ;  O,  $137.60;  P,$6.70;  Q,  $199.60 
Eailroad  Company,  $895  ;  S,  $9.60  ;  T,  $3.50  ;  U,  $266.75 
y,  $176.25  ;  and  W,  $2.75.     What  was  the  amount  ? 

12.  A  housekeeper  uses  .375  of  a  pound  of  sugar  to  a  jar 
of  fruit.  If  she  has  25  pounds  of  sugar,  how  many  jars 
can  she  put  up,  and  how  much  sugar  will  she  have  left  ? 


140  SECOND    BOOK  IN  ARITHMETIC. 

IS.  A  farmer  raised  823.3  bushels  of  oats  on  15  acres, 
and  466.45  bushels  on  7  acres.  What  was  the  average 
yield  per  acre  from  the  two  pieces  of  land  ? 

i^.  25  miles  of  a  railroad  57.36  miles  long  cost  $13,758.40 
per  mile,  17  miles  cost  $16,521.72  per  mile,  and  the  re- 
mainder cost  $18,125.12^  per  mile.  What  was  the  aver- 
age cost  per  mile  of  the  entire  road  ? 

15.  A  laborer  earns  $1.31J  per  day,  his  wife  $  .62J,  and 
each  of  3  children  $  .37^.  What  are  the  earnings  of  the 
family  per  week  ? 

16.  If  100  sheep  cost  $437.50,  what  is  the  price  per 
head  ?  • 

17.  How  many  bushels  of  onions  can  be  raised  on  3.27 
acres,  at  the  rate  of  415  bushels  to  the  acre  ? 

18.  How  many  tons  of  broom-corn  can  be  raised  from 
.85  of  an  acre,  if  1,876  tons  can  be  raised  from  one  acre  \ 

19.  The  owner  of  a  shcooner  sold  .3125  of  her  to  the 
captain.     What  part  of  the  vessel  did  he  still  own  ? 

W.  A  contractor  built  a  house  for  $3,725.  The  mate- 
rials cost  him  $2,641.37J,  and  he  paid  $1,796.50  for  labor. 
Did  he  make  or  lose  money,  and  how  much  ? 

^1.  What  is  the  cost  of  32.5  bushels  of  oats,  at  $  .56J? 

^2.  To-day  Mary  Morton  bought  of  me  15  yards  of 
dress  silk,  at  $1.75 ;  3  yards  of  satin,  at  $1.87i ;  ^  yards 
of  French  lace,  at  $.81^;  and  12  yards  of  gingham,  at 
$  .28.     Make  out  the  bill. 

23.  On  the  1st  day  of  May  last,  Andrew  Erwin  bought 
of  Potter  &  Co.,  of  Philadelphia,  6  pairs  of  calf  boots,  @ 
$4.50 ;  8  pairs  of  kip  boots,  @  $3.62|- ;  4  pairs  of  ladies' 
kid  boots,  @  $2.75 ;  and  12  pairs  of  ladies'  cloth  boots,  @ 
$2.12|-.     Make  out  and  receipt  the  bill. 


CHAPTER  III. 

PROPERTIES   OF  NUMBERS. 


SECTION  I. 

FACTORS  AND  DIVISORS. 

140.  The  integral  factors  of  a  number  are  those 
integers  of  which  the  number  is  the  product. 

One  integer  is  exactly  divisible  by  another  when  the  quotient  is  an 
integer. 

141,  A  composite  number  is  a  number  that  has 
other  integral  factors  besides  itself  and  1. 

14^.  A  prime  number  is  a  number  that  has  no 
integral  factors  besides  itself  and  1. 

a,  A  prifue  factor  is  a  factor  that  is  a  prime  number. 
&.  An  even  number  is  a  number  that  is  exactly  divisible 

by  2. 
c.  An  odd  number  is  a  number  that  is  not  exactly  divisi- 
ble by  2. 

Oral  Work.  —  1.  JSTame  all  the  composite  numbers 
between  30  and  75,  and  tell  why  they  are  composite. 

^.  Name  all  the  prime  numbers  from  1  to  50. 

3.  What  integers  less  than  the  number  20  are  even? 
Why? 

4^.  What  integers  between  30  and  50  are  odd?     Why? 

143.  Factoring  is  the  process  of  finding  the  integral 
factors  of  a  number. 

An  exact  divisor  of  a  number  is  any  integral  factor  of 
that  number. 

A.  What  are  the  integral  factors 
1.  Of  21?    I   2,  Of  35?    I   3,  Of  18?   I    Jf,  Of  29?   |    5,  Of  63? 


142 


SECOND   BOOK  IN  ARITHMETIC. 


What  numbers  are  exact  divisors 


6.  Of  15? 

7.  Of  27? 

8.  Of  35? 


9.  Of  24  ? 
10.  Of  56  ? 
ii.  Of  45  ? 


i^.  Of  11  ? 
i5.  Of  49  ? 
>4.  Of  95  ? 


15.  Of  64? 
i^.  Of  75  ? 
i7.  Of  108? 


What  are  the  prime  factors 

i<?.  Of8?     mOf36?    mOf50?    ^i.  Of72?    ^^.  Of  128? 

Any  number  is  exactly  divisible 
Hy  2f  if  it  is  an  even  number. 

By  3f  if  the  sum  of  its  digits  is  exactly  divisible  by  3. 
By  4,  if  the  number  expressed  by  its  two  right-hand  figures  is 

exactly  divisible  by  4. 
By  5,  if  its  right-hand  figure  is  5  or  0. 
By  O,  if  it  is  even,  and  the  sum  of  its  digits  is  exactly  divisible 

by  3. 
By  8,  if  the  number  expressed  by  its  three  right-hand  figures  is 

exactly  divisible  by  8. 

By  Of  if  the  sum  of  its  digits  is  exactly  divisible  by  9. 
By  10 f  if  its  right-hand  figure  is  0. 

^.  Of  the  twelve  numbers  in  the  margin, 
1.  Which  are  exactly  divisible  by  2  ? 


2.  By  3  i 

3.  By  4  ? 


^.  By  5  ? 

5.  By  6  ? 

6.  By  8  ? 


7.  By    9 

8.  By  10? 


78 

416 

656 

95 

360 

777 

168 

695 

1,260 

252 

423 

2,520 

Case  I.  Prime  factors. 

Written  Work. — Ex.  Find  the  prime  factors  of  210. 


210 


Process. 
105      35 


Explanation.  —  Since  210  is  an 

even  number,  I  divide  it  by  the 

prime  factor  2.     Since  the  sum 

of  the  digits  of  105  is  exactly  di-  ^  3  5         7 

visible  by  3, 1  divide  105  by  the 

prime  factor  3.     Since  the  right-hand  figure  of  35  is  5, 1  divide 

35  by  the  prime  factor  5,  and  obtain  the  prime  number  7. 
2,  8,  5,  and  7  are  the  prime  factors  required. 


PROPERTIES   OF  NUMBERS.— FACTORS,  ETC.    143 


144.  Rule  foe  Prime  Factoes. 
I.  Divide  the  given  member  iy  any  jprhne  factor, 
11,  If  the  quotient  is  a  com/posite  number^  divide  it  in 
lihe  manner ;  and  so  continue  to  divide^  till  the  quotient 
is  a  jprime  number. 

The  divisors  and  the  last  quotient  are  the  prime  factors. 
Problems. 


Find  the  prime  j  1. 
factors  of       ( 2. 


5.  729. 

6.  954. 


7.  2,838. 

8.  9,765. 


315.       S,  555. 
436.       Jf.  863. 

Case  II.  Common  Prime  Factors. 

Oral  Work, — 145,  "What  number  is  an  exact  divisor  of 


X   9  and  15? 

2.  16  and  28? 

3.  45  and  63  ? 
i.  19  and  72? 


5.  9,  12,  and  15? 

6.  18,  30,  and  42? 

7.  36,  84,  and  120? 

8.  27,  45,  and  72? 


9.  54,  36,  and  81  ? 

10.  $30,  $75,  and  $90? 
IJ.  $.27,  $.63,  and  $.81? 
12.  $50  and  $90  ? 


Process. 

90 

J^5 

15 

3 

120 

60 

20 

k 

2  JO 

106 

35 

7 

Written  Work.  —  Ex.  Find*  all  the  common  prime 
factors  of  90,  120,  and  210. 

Explanation. — 1  write  the  numbers 
in  columns,  as  for  addition,  and  di- 
vide successively  by  the  common 
prime  factors  2,  3,  and  5,  and  write 

the  quotients  in  columns  at  the  right  ^  ?  /r 

of  the  numbers  divided.  ^  o  o 

The  factors  2,  3,  and  5  are  the  common  prime  factors  required. 

Problems. 
Find  all  the  common  prime  factors 

1.  Of  36  and  80.     3.  Of  72  and  240.       5.  Of  45,  105,  and  180. 

2.  Of  64  and  96.     Jf.  Of  120  and  400.     6.  Of  52,  72,  and  148. 

Case  III.  Greatest  Common  Divisor. 

Oral  Work. — 146.  What  is  the  greatest  exact  divisor 


1.  Of  27  and  36? 

2.  Of  42  and  70  ? 


3.   Of  66  and  99  ? 
Jf.   Of  80  and  108? 


5.  Of  24,  40,  and  56  ? 

6.  Of  75,  45,  and  120? 


144 


SECOND    BOOK  IN  ARITHMETIC. 


147.  A  common  divisor  of  two  or  more  numbers 
is  any  factor  common  to  those  numbers. 

148.  The  greatest  co^nmon  divisor  of  two  or  more 
numbers  is  the  greatest  factor  common  to  those  numbers. 

1.  What  composite  number  is  the  greatest  common  di- 
visor of  24,  60,  and  96  ? 

^.  Of  what  prime  factors  is  this  greatest  common  di- 
visor the  product  ? 

3.  The  prime  factors  of  this  greatest  common  divisor 
are  also  prime  factors  of  how  many  of  the  numbers  24, 
60,  and  96  ? 

^.  The  product  of  any  two  or  more  numbers  is  the 
greatest  common  divisor  of  what  numbers  ? 

149.  Erinciple.  The  greatest  common  divisor  of  two 
or  more  numbers  is  the  product  of  all  their  common 
jprime  factors. 

Written  Work. — Ex.  Find  the  greatest  common  di- 
visor of  105,  525,  and  315. 

Explanation.  —  I  find  the 
common  prime  factors  of 
105,  525,  and  315  to  be  3, 
6,  and  7.  1  then  multiply 
these  common  prime  fac- 
tors together,  and  obtain 
105,  which  is  the  greatest 
common  divisor  of  the  given  numbers. 

Problems. 
Find  the  greatest  common  divisor 

6.  Of  105,  135,  and  180. 

7.  Of  288,  216,  and  504. 

8.  Of  280,  196,  and  112. 


Procesb. 

105 

35 

7 

1 

525 

175 

35 

5 

315 

105 

21 

3 

X    5   X    7 


105 


1,  Of  96  and  128. 

2.  Of  240  and  1,200. 

5,  Of  196  and  84. 
i.  Of  102  and  153. 

6.  Of  351  and  3,861. 


9,  Of  168,  280,  182,  and  252. 
10.  Of  2,835,  5,670,  and  4,455. 


PROPERTIES  OF  NUMBERS.-^FACTORS,  ETC,    145 

150.  EuLE  FOR  Greatest  Common  Divisor. 

I.  Divide  the  given  mimbers  hy  any  common  prime 
factor;  divide  the  residts  in  the  same  manner;  and  so 
continue^  till  results  are  obtained  that  have  no  common 
prime  factor, 

II.  Multiply  together  all  the  numbers  used  as  divisors. 

Problems. 
Find  the  greatest  common  divisor 

5.   Of  18,  27,  and  45. 


1.  Of  28  and  98. 

2.  Of  116  and  732. 

3.  Of  252  and  280. 
^.  Of  450  and  792. 


6.  Of  78,  90,  and  378. 

7.  Of  2,835,  2,455,  and  5,670. 

8.  Of  3,150,  4,050,  and  4,950. 


9.  What  is  the  length  of  the  longest  chain  that  will 
exactly  measure  the  length  and  width  of  a  piece  of  land 
which  is  160  rods  long  and  100  rods  wide  ? 

10.  A  farmer  draws  to  market  1,200  bushels  of  wheat, 
864  bushels  of  corn,  784  bushels  of  barley,  and  1,786  bush- 
els of  oats — each  kind  by  itself — in  loads  of  the  greatest 
possible  equal  number  of  bushels.  How  many  bushels 
does  he  draw  at  a  load  ?  How  many  loads  of  each  kind 
does  he  draw  ? 

11.  If  1,080  yards,  360  yards,  680  yards,  and  480  yards 
of  carpeting  are  laid  on  the  floors  of  rooms  of  equal  size 
in  a  hotel,  and  the  largest  size  possible  that  exactly  uses  all 
the  carpeting  of  each  kind,  how  much  carpeting  is  used 
for  a  room  ?     How  many  rooms  are  carpeted  ? 

12.  A  merchant  tailor  used  three  pieces  of  cloth  con- 
taining 95,  205,  and  380  yards  in  making  suits,  using  the 
same  amount  of  cloth  for  each  suit,  and  the  greatest 
amount  possible  without  leaving  remnants.  How  many 
suits  did  he  make  ? 

Q 


SECTIOTT   IL 

MULTIPLES. 

151.  A  multiple  is  any  integer  of  which  a  given 
integer  is  a  factor. 

Oral  Work* — i.  Name  some  number  that  is  a  multi- 
ple of  4. 

A  multiple  of  4  Is  any  number  of  which  4  is  an  integral  fac- 
tor ;  and  4  is  an  integral  factor  of  2  times  4  or  8,  3  times  4  or 
12,  4  times  4  or  16,  and  so  on. 

Name  three  numbers  )    2.  Of  7.    I    4-  Of  6.    I    6.  Of  15. 
that  are  multiples      )    S,  Of  5.    |    5.  Of  9.    |    7.  Of  40, 

Case  I.  Common  multiples. 

15^.  i.  Name  the  first  four  multiples  of  3. 
^.  Name  the  first  three  multiples  of  4. 
3.  Which  one  of  these  multiples  is  a  multiple  of  3  and  4? 
^.  What  number  is  a  multiple  of  4  and  6  ? 
A  multiple  of  4  and  6  is  a  number  of  -which  4  and  6  are  inte- 
gral factors ;  and  4  and  6  are  integral  factors  of  4  times  6  or  24. 

What  number  is  a  multiple 

5.  Of  3  and  8  ?       7.  Of  4  and  7  ?  9.  Of  3,  2,  and  11  ? 

6,  Of  2  and  5  ?       8,  Of  6  and  25  ?       10.  Of  4,  V,  and  5  ? 

153.  A  com^non  multiple  of  two  or  more  num- 
bers is  any  integer  of  which  each  of  the  given  numbers 
is  an  integral  factor. 

1.  What  are  the  prime  factors  of  12  ? 

^.  What  are  the  other  factors  of  12  ? 

5,  Then,  of  what  numbers  is  12  a  multiple  ? 
^.  Of  what  numbers  is  40  a  multiple  ? 

S, .  Of  what  numbers  is  21  a  multiple  ? 

6,  Find  a  common  multiple  of  6  and  10. 


PROPERTIES    OF  NUMBERS,— MULTIPLES.      147 


Find  a  common  multiple 


7.  Of  9  and  4. 

8.  Of  7  and  12. 


a  Of  6  and  3. 
m  Of  8  and  20. 


11.  Of  3,  5,  and  4. 
i^.  Of  8,  10,  and  7. 


13.  What  number  is  a  common  multiple  of  9, 10,  and  12  ? 

Written  Work. — Ex.  Find  a  common  multiple  of  16 

and  25. 

Explanation.-— I  multiply  16  and  25  to-  Process. 

gether,  and  obtain  their  common  multi-       i£>^QC_!nn 
pie,  400.  lb  X^O  -JfUU 

Problems. 
Find  a  common  multiple 


1.  Of  9  and  16. 

2.  Of  17  and  22. 


3.   Of  5,  12,  and  19. 
i.   Of  8,  13,  and  50. 


5.  Of  10, 11,  3,  and  31. 

6.  Of  6,  29,  and  67. 


Case  II.   Least  common  multiple. 

Oral  Work. — 154.  What  number  is  a  multiple,  and 
what  is  the  least  number  that  is  a  multiple,  of 


1.  2  and  6  ? 

2.  3  and  6  ? 

3.  4  and  6  ? 


Jf.  6  and  9  ? 

5.  4  and  10? 

6.  6  and  8  ? 


7.  2,  4,  and  12? 

8.  2,  6,  and  5  ? 
P.  3,  12,  and  10? 


10.  20  and  30? 

11.  10  and  25? 

12.  3,  4,  6,  and  8  ? 


155.  The  least  coinmon  multiple  of  two  or  more 
numbers  is  the  least  integer  of  which  each  of  the  given 
numbers  is  an  integral  factor. 

1.  What  number  is  a  common  multiple  of  8  and  12  ? 

2.  What  are  the  prime  factors  of  8  ?     Of  12  ? 

3.  What  number  is  a  common  multiple  of  all  the  prime 
factors  of  8  and  12  ? 

4^.  What  number  is  the  least  common  multiple  of  8 
and  12  ? 

150.  Principle.  The  least  comynon  multiple  of  two  or 
more  numhers  is  a  multiple  of  all  the  prime  factors  of 
eoAih  of  those  numbers^  and  of  no  other  prime  faxitors. 


148  SECOND    BOOK  IN  ARITHMETIC, 

Written  Work. — Ex.  Find  the  least  common  multi- 
ple of  20,  48,  and  56. 

Explanation.— I  first  separate  Process. 

each  of  the  numbers  into  its  6)  n  -^  o       o       r 

prime  factors.  ^^0  =  ^X^X0 

Then,  comparing  separately  the  1^8=^^x2x2x^x3 

prime  factors  of  20  and  48  66  =  2x2x2x7 

"''IV^^.lf^'JTl^,^      56x5x2x3=1,680 
— the  greatest  given  number,  ' 

— I  find  that  in  the  prime  fac- 
tors of  56  I  have  all  the  prime  factors  of  20  but  5,  and  all  the 
prime  factors  of  48  but  2  and  3 ;  and  I  write  tiiese  prime  factors 
and  56  in  a  line,  as  factors  of  the  required  least  common  multiple. 
Multiplying  these  factors  together,  I  obtain  1,680,  which  is  the 
least  common  multiple  of  the  given  numbers. 

Problems. 
What  is  the  least  common  multiple  of 


L  24  and  66  ? 
2,  15  and  100? 
S,  8,  12,  and  48? 
i.  6,  15,  and  90? 


5.  9,  21,27,  and  63? 

6.  32,  56,  96,  and  72? 

7.  36,  18,  and  24? 

8.  115,  30,  and  75? 


9.  54,  72,  and  126? 
10,  75,225,  and  400? 
IL  16,  81,  49,  and  25? 
12.  216,  132,  and  1,728? 


157.  Ex.  Find  the  least  common  multiple  of  6,  24, 


96,  15,  and  60. 

Explanation. — I  write  the  Process. 

numbers  in  a  line,  in  or-    v^        -v^        ^  #         /,  ^         q  ^ 
der,  from  least  to  greatest.       ^       "^^        ^^^       ^^         ^^ 
Since  6  and  15  are  factors      6  0  =  2x2x3x5 
of  60,  and  24  is  a  factor  of      9  6  =  2  X  2  X  2  X  2  X  2  X  3 

96,  I  strike  out  the  num-      • 

bers  6, 15,  and  24,  and  find      96  X  5  ■=  1^,80 

the  least  common  multiple 

of  the  remaining  numbers  60  and  96. 

This  least  common  multiple,  480,  is  the  least  common  multiple  of 
all  the  given  numbers. 

Wlien  any  of  the  given  numbers  are  factors  of  other  given  numbers, 
omit  those  numbers  tliat  are  factors,and  ftnid  tJie  least  comnion  mul- 
tiple of  the  oVier  nuTribers. 


PROPERTIES    OF  NUMBERS.— MULTIPLES.     149 

158.  Rules  foe  Multiples. 

I.  To  find  a  common  multiple:— 
Multiply  the  numbers  together. 

II.  To  find  the  least  common  miiltiple:— 

1.  Separate  the  miriibers  into  their  prime  factors. 

2.  Multiply  together  the  largest  given  number  and  those 
prime  factors  of  the  other  given  numbers  not  found 
among  the  factors  of  this  largest  number. 

Problems. 


Find  a  common  multiple 

1.  Of  50  and  81. 

2.  Of  3,  8,  and  70. 
8.   Of  20  and  67. 
Jf.   Of  135  and  318. 


Find  the  least  common  multiple 

5.  Of  the  first  nine  integers. 

6.  Of  5,  51,  10,  34,  and  85. 

7.  Of9,60,45,72,15,35,18,andl2. 

8.  Of  21,35,7, 100,15,  28,  and  125. 

9.  What  is  the  least  sum  of  money  with  whicli  I  can 
purchase  melons  at  45  cents  each,  or  plums  at  27  cents 
a  peck,  or  peaches  at  90  cents  a  basket,  or  oranges  at  36 
cents  a  dozen? 

10.  D  can  pick  40  bushels  of  apples  in  a  day,E  48  bush- 
els, and  F  64  bushels.  What  is  the  least  number  of  bush- 
els that  will  make  a  number  of  full  days'  work  for  any  one 
of  the  three  men  ? 

11.  A  can  walk  round  a  mile  course  in  12  minutes,  B 
in  16  minutes,  and  C  in  20  minutes.  If  they  all  start  to- 
gether, and  walk  till  they  are  again  together,  how  many 
minutes  will  each  walk?  How  many  miles  will  each 
walk? 

12.  What  is  the  least  number  of  gallons  of  petroleum 
that  will  fill,  some  exact  number  of  times,  any  one  of 
five  casks  that  hold  respectively  34,  54,  68, 108,  and  238 
gallons  ? 


SECTION  III. 

CANCELLATION. 

Oral  WorJc. — 159.  What  factor  will  remain,  if  I  re- 
move or  strike  out 

L  From  21  the  factor  7  ?    The  factor  3  ? 

^.  From  18  the  factor  3  ?   The  factor  6  ?  The  factor  2  ? 

S,  What  is  the  quotient  of  75  divided  by  15  ? 

^.  Of  i  of  75  divided  by  i  of  15  ? 

5.  Of  I  of  75  divided  by  |  of  15  ? 

6.  What  factors  are  common  to  15  and  75  ? 

7.  If  I  strike  out  from  75  all  the  factors  of  15,  what 
factors  will  remain  ? 


Divide 


(5.  4  X  6  by  4  X  3 
9,  8  X  7  by  2  X  T, 
10.  5X12  by  10X6. 
111.  9X8  by  3x12. 


12.  6  X  9  X  8  by  8  X  6  X  3. 

13.  4xl0x7xl2by4xl0xl2. 
H.  3x9  or  27  by  2  X  9  or  18. 
15.  7  X  2  X  6  by  2  X  7. 

16.  If  I  omit  all  the  common  factors  from  60  (the  divi- 
dend), and  24  (the  divisor),  what  factor  of  the  dividend  will 
remain  ?     What  factor  of  the  divisor  will  remain  ? 

17.  If  from  the  number  28  I  strike  out  the  factor  7, 
what  is  the  result?  What  has  been  done  to  the  num- 
ber? 

18.  The  dividend  is  64  and  the  divisor  is  16 ;  what  is 
the  quotient  ?  Strike  out  the  common  factor  8  from  divi- 
dend and  divisor ;  what  is  the  quotient  ? 

160.  Cancellation  is  the  process  of  omitting  or  strik- 
ing out  equal  factors  from  the  dividend  and  divisor. 

The  solution  of  problems  requiring  both  multiplication  and  divis- 
ion, may  often  be  shortened  by  cancellation. 


PROPERTIES   OF  NUMBERS.— CANCELLATION.     151 


161.  Principle  I.  Cancelling  a  factor  from  a  num- 
ber divides  the  number  by  that  factor, 

Pkinciple  II.  Cancelling  a  common  factxrr  from  divi- 
dend and  divisor  does  not  change  the  value  of  the  quotient. 

Written  TFoi^A?.— Ex.  1.  Divide  6  x  9  x  20  by  6  x  4. 

First  Process. 

5 


Explanation. — In  divi- 
dend and  divisor  are 
the  common  factors  6 
and  4  (20  =  5x4).  I 
cancel  these  factors, 
and  I  have  remaining 
the  factors  1,  9,  and  5 


1       1 


of  the  dividend,  and  1  and  1  of  the  divisor. 
I  multiply  together  the  remaining  factors  of  the  dividend,  and 

also  the  remaining  factors  of  the 

divisor,  and  obtain  45  for  a  new 

dividend  and  1  for  a  new  divisor; 

and   completing   the  division,  I 

have  45,  the  quotient  required. 
The  work  is  commonly  written  as 

shown  in  the  second  process. 


Second  Process. 

5 


Ex.  2.  Divide  16x21 
VX8. 

Process, 
^        S 
>^x^^^    6 


by 


Ex.  3.  Divide  8x25x45  by 
15x24x2. 

Process. 

:m.x  :^x  2    ^        ^ 

a.  Cancelling  any  factor  or  number  leaves  1  in  the  place  of  the 
factor  or  number  cancelled. 

h.  When  all  tlie  other  factors  of  either  dimdend  or  divisor  are  can- 
celled, write  the  1.    In  all  other  cases  omit  it. 

Pkoblems. 


1,  72  by  3  X  8. 
Divide  \  2.  5x5x5  by  25. 
S.  10x12  by  5x6. 


Jf,   14x16x18  by  7x8x9. 

5.  240x99  by  80x33. 

6,  20x32x44  by  5X6X11. 


152  SECOND    BOOK   IN   ARITHMETIC, 

7,  What  is  the  quotient  of  35  x  17  divided  by  5  x  7  ? 

8,  What  is  the  quotient  of  8  x  5  x  3  x  6-^9.x  10  x  4 ? 

9,  What  is  the  quotient  of  7  X  12  x  6  x  16-f-72  x  32  ? 

10  n  [2  13 

Divide       975  1365  5x28x5  5x9x5x7 

by       13x15  13x15  175  175 

14  [3  16 

Divide  200  X  24  X  36  7800  27  X  55  X  96x84 

by  144x60  390x4x3  33x132x28x15 

17,  The  factors  of  a  dividend  are  4,  15,  and  27 ;  and  of 
a  divisor,  3,  3,  4,  and  45.     What  is  the  quotient  ? 

18,  The  dividend  is  24x9x15x21,  and  the  divisor  is 
6  X  7  X  36  X 108.     What  is  the  quotient  ? 

19,  How  much  coffee,  at  30  cents  a  pound,  must  be  given 
in  exchange  for  6  pounds  of  butter,  at  40  cents  a  pound  ? 

W.  A  farmer  exchanged  360  sheep,  valued  at  $2.75  per 
head,  for  cows  at  $45  each.    How  many  cows  did  he  buy? 

^1,  How  many  bushels  of  turnips,  at  $  .25  per  bushel, 
will  pay  for  36  pounds  of  tea  at  $  .75  per  pound  ? 

^^.  If  a  boat  sails  24  miles  in  4  hours,  how  many  miles 
will  it  sail  in  2  x  8  hours  ? 

^3.  How  many  tons  of  hay  at  $20  a  ton,  are  worth  as 
much  as  60  bushels  of  wheat  at  $2  a  bushel  ? 

^^.  If  a  laborer  can  earn  $64  in  30  days,  how  much  can 
he  earn  in  35  days  ? 

^5.  If  12  men  can  do  a  piece  of  work  in  30  days,  in 
what  time  can  9  men  do  the  same  work  ? 

^6,  If  12  men  can  do  a  piece  of  work  in  30  days,  work- 
ing 10  hours  a  day,  in  what  time  can  15  men  do  the  same, 
working  8  hours  a  day  ?     ^ 

NoTB.~For  outlines  of  properties  of  numbers  for  review,  see  page  271. 


CHAPTER   IV. 

FRACTIONS. 


SECTION  I. 

NOTATION  AND  NUMERATION. 


1  OS.  1.  What  is  the  name  of  one  of  the  parts,  when 
any  thing  or  a  one  is  divided  into  two  equal  parts  % 


4.  Into  eight  equal  parts  ? 

5.  Into  ten  equal  parts  ? 


2.  Into  three  equal  parts  ? 
S.  Into  five  equal  parts? 

6.  Into  any  number  of  equal  parts  ?   . 

Any  thing  or  a  one  is  how  many 

7.  Halves?!  9,  Eighths?  iii.  Sixths?  \1S.  Fourths?  \15.  Tenths? 

8.  Fifths?  \lO.  Thirds?    \l2,  Ninths?|i4.  Sevenths? I i(5.  Twelfths? 

"When  any  thing  or  a  one  is  divided  into  nine  eqnal  parts, 
17.  What  is  the  name  of  the  parts  ? 


18.  Of  one  part  ? 

19.  Of  four  parts  \ 


20.  Of  two  parts? 

21.  Of  five  parts? 


22.  Of  seven  parts  ? 

23.  Of  three  parts  ? 


163.  Fractional  parts  are  parts  obtained  by  divid- 
ing any  thing  or  a  one  into  any  number  of  equal  parts. 

Halves,  fifths,  eighths,  twelfths  are  fractional  parts. 

164.  A  fractional  unit  is  one  of  the  equal  frac- 
tional parts  into  which  a  thing  or  the  unit  1  is  divided. 

1  half,  1  fifth,  1  eighth,  1  twelfth  are  fractional  units. 

165.  K  fraction  is  a  number  that  expresses  one  or 
more  fractional  units. 

1  sixth,  5  eighths,  9  tenths  are  fractions. 
A  fraction  is  expressed  by  two  numbers,  written  one 
under  the  other,  with  a  horizontal  line  between  them,  thus  - 
1  eighth,  ^ ;  3  eighths,  f ;  1  fifteenth)  1^ ;  11  fifteenths,  \\. 

G2 


154 


SECOND    BOOK  IN  ARITHMETIC, 


1,  What  numbers  express  the  fraction  five  eighths  ? 


What  is 

5.  The  fractional  unit  ? 

6.  The  number  of  fractional  units? 

7.  The  value  of  the  fraction  ? 


AYliich  number  determines 
2.  The  size  of  the  parts  ? 
S.  The  number  of  the  parts  ? 
U,  The  value  of  one  part? 

166.  The  terms  of  a  fraction  are  the  two  numbers 
used  to^xpress  it. 

167.  The  denominator  is  that  term  which  names 
the  parts  expressed  by  the  fraction.  It  is  written  helow 
the  horizontal  line. 

168.  The  numerator  is  that  term  which  numbers 
the  parts  expressed  by  the  fraction.  It  is  written  ahc/oe 
the  horizontal  line. 

a.  The  terms  of  the  fraction  4  are  4  and  5 ;  the  denominator  is  5 ; 
and  the  numerator  is  4. 

h.  The  denominator  shows  the  size  of  the  parts ;  and  the  numera- 
tor shows  the  number  of  parts  expressed  by  the  fraction. 

1 69.  The  numerator  of  any  fraction  is  a  dividend,  the 
denominator  is  a  divisor,  and  the  value  expressed  by  the 
fraction  is  the  quotient. 

•^  expresses  the  quotient  of  9  divided  by  16. 
A.  Read  each  of  the  fractions  given  below,  and  name 
i.  The  terras;  J^.  The  fractional  unit;   6,  The  dividend; 

2,  The  numerator;      5.  The  number  of  frac-   7.  The  divisor;  and 
S,  The  denominator;  tional  units;  8,  The  quotient. 

I      i      tV      ^      ii      ^      ¥      « 

J5.  The  fraction  f  expresses  2  of  the  3  equal  parts  into 
which  1  is  divided;  or  1  of  the  3  equal  parts  into  which 
2  is  divided. 


What  is  expressed 

i.  By  the  fraction  |  ?  5.  By  |  ? 

2,  By  the  number  |  ?  ^.  By  |^  i 


What  do  you  understand 

5.  By  3^?  1  ^-  By  A  of  a  pound? 

6.  Byi?    U.  By -^  of  a  dozen? 


FRACTIONS.— NOTATIOJSf  AND    NUMERATION     155 


C  1,  Wliicli  of  tlie  fractions  in  the  margin 
express  a  value  equal  to  1  ?    Why  ? 
^.  Which  express  less  than  1  ?    Why  1 
3,  Which  express  more  than  1  ?    Why  ? 


1  5 


Iff 


_7^     1. 5.       236 
1^     16      1,728 


236 
15T 


I  ¥ 


1  f'O,  A  proper  fraction  is  a  fraction  that  expresses 
less  than  1. 

171,  An  improper  fraction  is  a  fraction  that  ex- 
presses 1  or  more  than  1. 

a,  f ,  ^  are  proper  fractions ;  f ,  ^  are  improper  fractions. 
6,  The  numerator  of  a  proper  fraction  is  alicays  a  less  number 
than  the  denominator;  the  numerator  of  an  improper  fraction  is 
never  a  less  number  than  the  denominator, 

1 7S.  A  7nixed  number  is  a  number  that  is  expressed 
by  an  integer  and  a  decimal,  or  by  an  integer  and  a  fraction. 

a.  17.4,  3.65;  12|,  5y^^  are  mixed  numbers, 

b.  In  reading  a  mixed  number,  read  and  between  the  integer  and 
the  fraction, 

c.  Decimal  units,  as  well  as  fractional  units,  are  equal  parts  of  a 
thing,  or  of  the  unit  1.     Hence, 

1.  Decimal  parts  are  fractional  parts;  and 

2,  Decimals  are  also  called  decimal  fractions, 

1,  Which  of  the  numbers  in  the  margin  are     - 

proper  fractions,  and  why  ? 
^.  Which  are  improper  fractions,  and  why  ? 
3.  Which  are  mixed  numbers,  and  why  ? 


i 


f 


T¥ 


f 


A 


^ 


What  is 
the  unit  ^ 


Jf..  Of  315? 

5.  Of  $89  ? 

6.  Ofi? 

7.  Of  4? 


12.  Of  ^  of  an  acre  ? 

13.  Of  i  of  a  yard  ? 
IJf.  Of  3i  dozen  ? 


8.  Of  37  bushels? 

9.  Of  45  pounds? 

10.  Of  f  of  a  bushel  \ 

11,  Of  I  of  a  pound  ?  [  16.  Of  1 6f  miles  ? 

If^.  The  tmit  of  a  fraction  is  one^  of  the  kind 
expressed  by  the  fraction. 


What  is  the  unit  of  each  of 
the  numbers  in  the  maro'ln  ? 


28    iV  A  M      H     3f  weeks. 
413    i    I  1%  15-^  18f  gallons. 


156  8EC0ND    BOOK   IN   ARITHMETIC. 

174.  To  analyze  a  fraction  is  to  name  and  de- 
scribe all  its  parts,  its  kind,  and  its  value. 
Ex.  Analyze  the  fraction  f  of  a  mile. 

Analysis. — The  terms  of  this  fraction  are  3  and  4,  because  they 
are  the  two  numbers  used  to  express  it;  the  denominator  is  4, 
because  it  names  the  parts  expressed  by  the  fraction ;  and  the 
mim^rator  is  3,  because  it  numbers  the  parts  expressed  by  the 
fraction. 

The  unit  of  the  fraction  is  1  mile,  because  it  is  the  one  divided  to 
form  the  fraction ;  and  the  fractional  unit  is  i,  because  it  is  1 
of  the  4  equal  parts  into  which  1  mile  is  divided. 

The  fraction  is  proper,  because  it  expresses  less  than  1 ;  and  its 
xalue  is  f  of  a  mile,  because  f  expresses  the  quotient  of  the  nu- 
merator 3  divided  by  the  denominator  4. 


Analyze      j  i.  4-  of  a  wfefek. 
the  fractions  (  ^.  A  of  a  foot. 


I  Si. 


5.  f. 
6.^, 


7.  ^. 


SECTION   II. 

REDUCTIONS. 
Case  I.  Fractions  to  lowest  terms. 

Oral  Work. — 175.  1.  1  is  how  many  fifths?    How 
many  fifteenths?     How  many  sixtieths? 

^.  f^  are  how  many  fifteenths?     How  many  fifths? 
How  many  ones? 

3,  How  is  the  fraction  -f^  reduced  to  halves  ? 

Jf.,  How  is  the  fraction  -^-^  reduced  to  sixtieths?     To 
thirtieths  ?     To  fifteenths  ? 

6,  Dividing  the  terms  of  a  fraction  by  any  number,  has 
what  effect  upon  the  value  of  the  fraction  ? 
Which  expresses  the  greater  value, 

^.  iorf?      I      7.  for  A?      |      <^.  |^,  ^f ,  or  |  ? 

176.  A  f faction  is  in  its  lowest  terins^  when  its 
terms  have  no  common  factor. 


FRA  CTIONS.—RED  UCTIONS. 


157 


i.  Reduce  the  fraction  -/^  to  its  lowest  terms. 

I  divide  the  terms  8  and  28  by  the  common  factor  4,  and  ob- 
tain ^ ;  and,  since  the  terms  2  and  7  liave  no  common  factor,  the 
lowest  terms  of  the  fraction  ^^  are  |-. 


Reduce  to 
lowest  terms 


14- 


4-  A- 

6.  -H- 

5.  If 

5.  H- 

7.  *f- 

9.  If. 

12.  yy-  of  a  lemon  are  how  many  eighths  of  a  lemon? 
How  many  fourths?     How  many  halves? 

13.  Reduce  the  fractions  %%  and  $i|  to  lowest  terms. 
IJf.  James  has  f  of  an  apple,  and  Charles  has  -f^.   "Which 

has  the  greater  fraction  ? 

16.  Is  %^-Q  more  or  less  than  $/q-,  and  how  much  ? 

16.  In  five  successive  Vv^eeks  a  family  uses  -!-§-,  |-|,  -|,  ^-, 
and  ^  oi  2i  bushel  of  apples.  In  which  week  do  they 
use  the  most  ? 


177,  Principle  I.  The  value  of 
changed^  iy  dividing  its  terms  hy  any 

Written  Work. — Ex.  Reduce  l^f 

Explanation.— I  first  divide  105  and  175 
— the  terms  of  the  given  fraction — by  the 
common  factor  5,  and  obtain  f  1.  I  then 
divide  21  and  35 — the  terms  of  this  frac- 
tion— by  the  common  factor  7,  and  obtain 
f .  Since  the  terms  3  and  5  have  no  com- 
mon factor,  the  lowest  terms  of  the  frac- 
tion i^f  are  f . 

The  result  may  be  obtained  at  one  division, 
105  and  175  by  35,  their  greatest  common 

Peoblems. 
"What  are  the  lowest  terms  of 


a  fraction  is  not 
common  factor. 

to  lowest  terms. 
First  Process. 

105-^5  _3J.-^7  _S 
175^5  ~  55-f-7~5 

Second  Process. 


101^1^5  . 


by  dividing  the  terms 
factor. 


3.  M? 


•4.  It? 
5.  ft? 

e.  H? 


if? 


7. 

9- 1%? 


10.  J^? 

11.  ^? 

12.  -1^? 


IS.  ,%? 

ijf.  ,J4I? 

15-  ■ms'- 


158 


SECOND    BOOK  IN  ARITHMETIC. 


Reduce  to 
lowest  terms 


16.  \^^  of  a  yard. 
1^'  Mfl  of  a  day. 
18.  $3-Vjand$-j^. 


i^'  Um  of  a  pound. 
21.  |g]|P  of  a  ton. 


Case  II.  Fractions  to  other  fractions  having  given 
denominators. 

Oral  Work. — 178.  What  fraction  expresses  the  re- 
sult, when  each  of  2  halves  is  divided  into 


1.  2  equal  parts?     I     S.  7  equal  parts? 

2.  3  equal  parts?     |     4.  5  equal  parts? 

Multiply  the  terms  of  the  fraction 


5. 
6. 


10  equal  parts? 
8  equal  parts? 


I  l>y  3. 
§by4. 


9.^ 
10.  i  by  6, 


^by  8. 


11.  {%  by  9. 

12.  U  by  5. 


13. 
11 


if  by  10. 
f*by7. 


By  what  number  must  you  multiply  the  terms,  to  change 


15.  "I  to  twentieths? 


16. 


4  to  forty-eighths? 


17.  ■?■  and  ^  to  thirty-fifths? 


18.  -g-  and  -yy  to  ninety-ninths? 
To  reduce  a  fraction  to  another  fraction  having  a  larger  de- 
nominator : — Divide  the  required  denominator  hy  the  given 
denominator^  and  multiply  the  terms  of  the  given  fraction 
by  the  quotient, 

19.  Reduce  the  fraction  -^  to  twenty-eighths. 

7,  the  denominator  of  ^,  is  contained  in  28,  the  required  de- 
nominator, 4  times ;  hence,  I  multiply  the  terms  of  f  by  4,  and 
obtain  ■^. 

W.  Eeduce  A  to  30ths.     To  60ths.     To  120ths. 

21.  ^  to  fourths.     To  16ths.  2J^.  f  and  \  to  54tlis. 

22.  -I  to  fortieths.    To  64ths.  25.  ^  and  ^V  to  60tlis. 

23.  -f-  to  forty-seconds. 

'27. 
\28. 
\29. 


Which  is  the 
greater  number, 


26.  -J,  -J,  and  I  to  12ths. 
•|  of  an  hour,  or  ^\  of  an  hour? 
-^  of  a  bushel,  or  f  of  a  bushel  ? 
-^-^  of  an  acre,  or  f\j-  of  an  acre  ? 


1 79.  Principle  II.  The  value  expressed  hy  a  fraction 
is  not  changed,  hy  multiplying  its  terms  hy  any  number. 


FRACT10N8.~RED  UCT10N8.  159 

Written  Work. — Ex.  Eeduce  the  fraction  |  to  125tlis. 
Explanation.  —  Dividing    125,   the    re- 
quired denominator,  by  S,  the  given  de-  Process. 
nominator,  I  obtain  25,  the  number  by  -   ^    ^       ^  _, 
which  the  terms  of  the  fraction  f  must  1  /j  o  -r-  o  ^=^  Jq  o 
be  multiplied  to  reduce  it  to  12oths.     I  _ 
then  multiply  the  terms  3  and  5  by  25,  |x  ^^- J|_ 
and  obtain  ^^^,  the  required  result. 

Pkoblems. 
1,  Keduce  |-  to  a  fraction  having  468  for  a  denominator. 
^.  Seven  eighths  are  how  many  one  hundred  sixtieths  ? 


i  S.  ^^  to  eooths. 

Reduce  \  Jf.  -i^io  275ths. 

(  5,  -J-  to  QOOths. 


P.  ||-to801sts. 
ia|i|-toll84ths. 
ii.  3%-tol'728ths. 


a|-to297ths. 
7.  y\to285ths. 
<5.^^toll52ds. 

1^,  A  has  $1^,  B  has  $-|,  and  C  has  $f .  Which  of  the 
three  persons  has  the  most  money  ? 

13.  A  grocer  has  fV  ^^  ^  barrel  of  pulverized  sugar,  ^ 
of  a  barrel  of  granulated,  and  y^^^-  of  a  barrel  of  crushed. 
Of  which  kind  has  he  the  most  ? 

Case  III.  Dissimilar  fractions  to  similar  fractions. 

1 80,  Similar  fractions  are  fractions  that  have  the 
same  fractional  unit. 

181,  Dissimilar  fractions  are  fractions  that  have 
different  fractional  units. 

a,  f ,  f  are  similar  fractions ;  f ,  f ,  f,  L  i  are  dissimilar  fractions. 
&.  Similar  fractions  have  like  denominators. 
c.  A  common  denominator  is  the  denominator  of  each  of 
two  or  more  similar  fractions. 

Oral  Work. — 1,  Which  of  the  fractions 
in  the  margin  are  similar  fractions  ?     Why  ? 

2,  Which  are  dissimilar  fractions  ?    Why  ? 

3.  Which  have  a  common  denominator  ? 


i     JL  3       2 

27     7?  ¥>    ¥• 

h  i,  h  f  • 

4       6  3       3 

¥5    T?  ¥?    T* 

T?  TTJ  "S"?  TO"* 


^.  Which  can  be  reduced  to  eighteenths  ?    Why  ? 


160 


SECOND    BOOK  IN  ARITHMETIC, 


5.  The  denominators  of  fractions  that  can  be  reduced 
to  fifteenths,  must  be  factors  of  what  number  ? 

What  fractions  can  be  reduced 

6\  To  28ths  ?   Why  ?  |  7.  To  40ths  ?   Why  ?  |  <^.  To  48ths  ?   Why  ? 

What  similar  fractions  are  equal  in  value 

9.  To  I  and  f?  1 10,  Tofand^?  1 11,  To^and^?  1 12,  To  J,  f  and  |? 

The  product  of  all  the  denominators  of  two  or  more  frac- 
tions is  a  common  denominator  of  the  fractions.     Hence, 

182,  A  common  denominator  of  two  or  more  similar 
fractions  is  a  common  multiple  of  all  their  denominators, 

1.  Reduce  \  and  \  to  similar  fractions. 

Since  the  given  denominators  7  and  9  are  factors  of  63,  sev- 
enths and  ninths  can  be  reduced  to  sixty-thirds. 

Multiplying  the  terms  of  ^  by  9,  and  the  terms  of  J^  by  7,  I 
obtain  the  similar  fractions  ^  and  -g-^. 

Reduce  to  similar  fractions 


2.  The  fractions  \  and  f. 
S,  The  fractions  f  and  -^-^, 


4.  +  and  J. 

5.  \,  \,  and  f 


^.  h  i  and  f 
r.  I,  I,  and  f. 


Written  Work.—l  83,  Ex.  Reduce  |, 
^,  and  f  to  similar  fractions. 

Explanation.  —Since  3x2x9  =  54,  thirds,  halves, 
and  ninths  can  be  reduced  to  fifty-fourths.  I 
therefore  multiply  the  terms  of  each  fraction 
by  the  denominators  of  the  other  fractions. 

The  results,  ff ,  ||,  and  fj  are  similar  fractions. 

Problems. 


Process. 


2^X2X9  _ 
3X2X9 ' 


1X3X9  _ 
2X3X9 ' 


36 
''5k 


^27_ 
'  5U 


5X3X2  __50 
9X3X2        5k 


1,  Reduce  to  similar  fractions  ^,  |-,  and  -|. 


6.  I,  ^,  and  -^V 
^•iii.A.andi. 

^.A,iA,f,iandA. 


^.fand^V  4.i,|,l  andf. 

S,  -^  and  i|.  5,  I,  4,  -I,  and  ^. 

9.  Reduce  f,  f,  J^,  and  f  to   fractions    of  equal  value 
having  a  common  denominator. 


FRA  CTI0XS.-REDUCTI0N-8,  161 

10,  The  denominators  of  several  dissimilar  fractions  are 
13,  3,  10,  5,  and  12.  What  is  the  denominator  of  the 
similar  fractions  of  equal  value  ? 

11.  Keduce  |,  y^^,  f ,  |,  and  |  to  fractions  having  a  com- 
mon fractional  unit. 

Case  IV.  Dissimilar  fractions  to  least  similar  frac- 
tions. 

184.  Least  similar  fractions  are  fractions  that 
have  the  greatest  common  fractional  unit  possible. 

The  least  common  denominator  is  the  denominator  of 
least  similar  fractions. 

Oral  Work. — What  number  is  a  common  denominator 


1.  Of  land  1? 

2,  Of  i-and^? 


Of - 


6.  Of  I!  I!  and  I? 


S.  Of  land  ^? 
Jf.  Of  land  i? 

"What  number  is  a  common  denominator,  and  what  num- 
ber is  the  least  common  denominator 


7.  Of  i-andi? 

8.  Of  I  and  I? 


9.  Of  i  and 


10  ' 


.2 


11.  Ofi,i,  and^i,-? 

12.  Ofi,TV,andyV? 


10.  Offand-I 

185.  The  least  common  denominator  of  two  or  more 
similar  fractions  is  the  least  common  multiple  of  all 
their  denominators. 
By  what  number  must  you  multiply  the  terms 

1.  Of  \  and  yV?  ^^  reduce  the  fractions  to  24th8  ? 

2.  Of  -|,  yV?  ^^^  tV>  ^^  reduce  the  fractions  to  36ths  ? 

S.  Of  Y^-g-,  |- ,  ^^o",  and  -i ,  to  reduce  the  fractions  to  60ths  ? 
^.  Of  f ,  y\,  and  W^  to  reduce  to  least  similar  fractions  ? 

5.  Of  y\,  W^  y%,  and  -J-,  to  reduce  to  least  similar  fractions  ? 

6.  Eeduce  f  and  -|  to  least  similar  fractions. 

Since  12  is  the  least  common  multiple  of  the  denominators  4 
and  6,  the  fractions  can  be  reduced  to  twelfths.  Since  12  equals 
4  times  3  or  6  times  2, 1  reduce  |  to  twelfths  by  multiplying  its 
terms  by  3,  and  f  to  twelfths  by  multiplying  its  terms  by  2  ; 
and  I  obtain  the  least  similar  fractions  ^  and  ^. 


162 


SECOND   BOOK  IN  ARITHMETIC. 


Eeduce  to  least  similar  fractions 


7.  Tlie  fractions  ^  and  3^. 

8.  The  fractions  f  and  -^. 


9.  f  and  f       I  11.  -gV,  i^,  and  ^. 
10.  -^  and  ^V  I  ^^.  f ,  iV,  and  H- 


Written  Work. — 186.  Ex.  Eeduce  the  dissimilar 
fractions  f ,  ^,  and  ^  to  least  similar  fractions. 


Explanation. — I  first 
find  the  least  com- 
mon multiple  of  the 
given  denominators 
—9, 15, 18— to  be  90. 

I  next  divide  this  90  by 
the  given  denomina- 
tors 9, 15, 18,  and  ob- 
tain 10,  6,  5. 

I  then  multiply  the 
terms  of  |  by  10,  the 
terms  of  ^^  by  6,  and 
the  terms  of  }  J  by  5  ; 


9 

3 

1 

15 

5 

5 

18 

6 

9. 

Process. 


3x3x5  x2  =  90 

90^  9=10 
90-^15=  6 
90^18=    6 


Hence,    -»  — »  —  : 
9    15    18 


U  XIO  _  UO 
9  X10~  90 

2_X  6   _1S 
15X  6   ~  90 

1J.X  5   _55 
18X  5   ~  90 

,1*0   11    55^ 
■  90'  90^  90' 


Reduce  to  least 
similar  fractions 


5.  if,  I,  and  |. 

6.  f ,  i,  and  H. 

^.  A,  H,  i  and  iJ. 
^'  i  H,  h  H,  and  i 


and  obtain  |J,  Jf ,  and  f  J,  the  least  similar  fractions  required. 
Problems. 

fi.  T^andA. 
2.  ^%  and  VV- 
S.  ^,  i,  and  f . 
U.  hh  andf. 
9,  Reduce  f,  ^,  -j^,  and  f  to  equivalent  fractions  hav- 
ing the  least  common  denominator. 

10,  What  is  the  fractional  unit  of  the  least  similar  frac- 
tions to  which  f ,  f ,  ^,  and  -^  can  be  reduced  ? 

Case  V.  Improper  fractions  to  integers  or  mixed 
numbers. 

Oral  Work. — 18*To  1.  ^-^  are  how  many  ones? 

Since  8  eighths  are  1,  59  eighths  are  as  many  I's  as  the 
times  8  eighths  are  contained  in  59  eighths.  1  eighth  is  con- 
tained in  59  eighths  59  times,  and  8  eighths  are  contained  in  59 
eighths  1  eighth  of  59  times,  which  is  7f  times. 


FRA  CTIONS.—RED  UCTI0N8, 


163 


How  many  I's 

2.  Arel2foniths? 
S.  Are  36  sixths? 


Jf.  Are  28  sevenths? 
5,  Are  24  thirds? 


6.  Are  54  ninths  ? 

7.  Are  55  fifths? 
10.  Are -3^? 


I  9.  Are-V-? 
Reduce  to  a  mixed  number 
i^.  The  fraction  ^~, 


15. 


The  fraction  ^-. 


8.  How  many  I's  are  ^  \ 
Reduce  to  an  integer 

11.  The  fraction  -S/. 

12.  The  fraction  ■^. 
IS.  The  fraction  -i^.  i6'.  The  fraction  ^^. 

17.  How  is  an  improper  fraction  reduced  to  an  integer 
or  a  mixed  number  ? 

When  the  denominator  is  a  factor  of  the  numerator,  the  'calue  of  the 
fraction  is  an  integer. 

Written  Work. — 188.  Ex.  Reduce  -%^-  to  an  inte- 
ger or  a  mixed  number. 

Explanation. — Since  the  numer-  Process. 

ator  of  a  fraction  is  a  dividend,       593  ^  i^  q  o  _^oq  ^  1 R  4rt 
and  the  denominator  is  a  divi-       ^^  ~  *  *^ 

sor,  I  divide  the  numerator  593 
by  the  denominator  32,  and  obtain  the  mixed  number  18^. 

Problems. 
Reduce  to  integers  or  mixed  numbers 


2     5|_8. 

5.  W- 

^.  w. 

IS,    35|lg^ 

17.  ^. 

2.  ^^. 

6.  mK 

10,  ^^^, 

11    ^¥^. 

IS,  H^. 

S.  -1^. 

7.  H^. 

11.  ^^. 

15,    12AA, 

19,  loj^. 

J,.  W. 

<?.  W. 

12.  ^fr-. 

16.  i|fi-. 

20.  ^i^. 

21.  153  bales  of  hay,  each  weighing  -i-  of  a  ton,  weigh 
how  many  tons  ? 

22.  In  1,760  baskets,  each  containing  -J-  of  a  bushel  of 
peaches,  are  how  many  bushels  ? 

How  many  dollars  are 

2S.  1,954  quarter-dollars?     |     2^.  $i^?     |     25.  $^3^? 


164 


SECOND    BOOK  IN  ARITHMETIC. 


Case  VI.  Integers  or  mixed  numbers  to  improper 
fractions. 

Oral  Work. — 180.  1.  5f  are  how  many  fourths? 

Since  1  is  4  fourths,  5  are  5  times  4  fourths,  or  20  fourths; 
and  5  j  are  20  fourths  plus  3  fourths,  which  are  23  fourths. 

^.  How  many  fourths  are  S^J-?    Are  20|?    Are  15  ? 

3.  Reduce  9  to  fifths.     ^  to  fifths.     20|  to  eighths. 


Reduce 


[  ^.  7  to  fifteenths. 
'  5.  4  to  twelfths. 


6,  8  to  fifths.     I  8.  5|  to  eighths. 

7.  7|  to  thirds.  |  9.  123iQ-to  tenths. 

10,  How  is  an  integer  reduced  to  an  improper  fraction  ? 

11,  A  mixed  number  to  an  improper  fraction  ? 


Written  Work. — 190.  Ex.  Reduce  27f  to  an  im- 
proper fraction. 

Explanation.— I  multiply 
5,  the  number  of  fifths  in 
1,  by  27,  and  obtain  135, 
the  number  of  fifths  in 
27.  To  this  result  I  add 
the  8  fifths  of  the  given 
number,  and  obtain  J-f^, 
the  improper  fraction  re- 
quired. 


First  Process.  | 

S 

fifths. 

27 

135 

fifths. 

3 

fifths. 

138 

fifths. 

Second  Process. 

5_ 

135+3  =  138 


Hence,  27f  =  ^P-, 
The  work  is  commonly  written  as  shown  in  the  second  process. 
Pkoblems. 


1.  18  is  how  many  ninths? 

2.  24^  are  how  many  fourths  ? 

3.  36'|-  are  how  many  eighths? 


Jf.  How  many  53ds  are  *l\^'i 

5.  How  many  37ths  is  37? 

6.  Reduce  11^  to  fifteenths. 


What  improper  fraction  is  equal 


7. 

Tol7i? 

8. 

To  27jV? 

9. 

To  24|? 

W. 

To  2^? 

11. 

To  U^\? 

12. 

To  Yo^V? 

13. 

To  17^? 

u. 

To  20^  ? 

15.  To  106^? 

16.  To87^3_? 

17.  To  5^? 

18.  To  21||? 


19.  To  31|? 

20.  To  33yV? 

21.  To  115^1^? 

22.  To  86^9^  ? 


FRA  CTIONS.—RED  UCTIONS.  165 

191.  EuLEs  FOR  Reductions  of  Fkactions. 

I.  A  fraction  to  lowest  terms. 

Cancel  all  the  factors  common  to  the  terms  of  the  fraction, 

II.  A  fraction  to  another  fraction  having   a  given 

denominator. 

Divide  the  given  denominator  hy  the  denominator  of  the 
fraction^  and  multijply  both  terms  of  the  fraction  hy  the 
quotient, 

III.  Dissimilar  fractions  to  similar  fractions. 

Multiply  the  terms  of  each  fraction  hy  the  denominators 
of  all  the  other  fractions, 

lY.  Dissimilar  fractions  to  least  similar  fractions. 

1.  For  the  least  common  denominator ; — Find  the  least 
common  multiple  of  all  the  denominators, 

2.  For  each  new  numerator ; — Divide  the  least  common 
midtiple  hy  the  denominator  of  each  fraction^  and  mxdti- 
ply  the  numerator  hy  the  quotient. 

Y.  An  improper  fraction  to  an  integer  or  a  mixed 
number. 

Divide  the  numerator  hy  the  denominator, 

YI.  An  integer  or  a  mixed  number  to  an  improper 
fraction. 

1.  Multiply  the  integer  hy  the  denominator ^  and  if 
there  he  a  numerator  add  it  to  the  product, 

2.  Write  this  result  for  the  numerator,,  and  the  given 
denominator  for  the  denominator  of  the  required  fraction. 

Rules  I  and  V  are  used  for  reducing  final  results  to  their  simplest 
forms;  rules  II,  III,  and  IV  are  used  for  preparing  numbers 
for  addition,  subtraction,  and  division ;  and  rule  VI  is  used  for 
preparing  numbers  for  multiplication  and  division. 


SECTION  III. 

ADDITION. 


Oral  Work.—l^^.  L  What  is  the  sum  of  |  and  |? 

^.  What  is  the  sum  of  \,  |,  and  f  ? 

S.  What  is  the  sum  of  -^^,  ^^,  -g^,  and  ^  ? 

^.  To  what  must  halves  be  added  ?    Ninths?   Sixteenths? 

5.  What  similar  fractions  equal  f  and  \  ?     What  is  their 
sum? 


What  is  the  sum  of 


e,  -I  and  4  ? 
7.  A  and  I? 


<^.  -I  and  4? 

P.  I  and-jij? 


Add 

ia  I  and  f 
ii.  A  and  f 


i<?.  2i  and  f 


TF^n  aTiy  o/"  ^Ae  'parts  are  mixed  numbers,  add  t?ie  fractions  first. 


Add 


(  U.  4-j^  and  f 
(  15.  5^  and  7J. 


i(?.  4,  2-1,  and  17. 
iP.  5|,  3|-,  and  4f. 


i<?.   Ie5f  and  8|. 
i7.  3|  and  6f 

^6?.  $-J-  for  com  and  $^  for  peas  is  how  much  for  both  ? 
2L  Esther  paid  $f  for  overshoes,  $|-  for  gloves,  and  $J 
for  handkerchiefs.     How  much  did  she  pay  for  all  ? 

22.  Eli  paid  $2f  for  a  calf,  and  sold  it  for  $j\  more 
than  it  cost  him.     For  how  much  did  he  sell  it  ? 

23.  If  I  burn  4.  of  a  ton  of  coal  one  month,  and  f  of  a 
ton  the  next,  how  much  do  I  burn  in  the  two  months  ? 

24.  Eoger  sawed  a  load  of  wood  in  1^3^  days,  and  split 
it  in  1^  days.     How  much  time  did  he  work  ? 

25.  My  orchard  contains  3|-  acres,  and  my  garden  1^ 
acres.    How  much  land  is  there  in  my  orchard  and  garden  ? 

26.  Harry  is  S-^^  years  old,  Carlos  is  6 J  years  older  than 
Harry,  and  Hattie  is  5^  years  older  than  Carlos.  How  old 
is  Hattie  ? 


FRA  CTIONS.--ADDITIOir. 


167 


^7.  Can  f  of  a  day  and  f  of  a  day  be  added  ?    Why  ? 
28,  Can  f  of  a  mile  and  |  of  a  dollar  be  added  ?    Why  ? 

Written  Work.— 19^.  Ex.  1.  What  is  the  sum  of 
i  i,  andf? 

First  Process. 


5'^  S~^  h         60  ~^  60'^  60  60    ' 


^  60 


Second  Process. 


EXPLANATIOIC.— 

Fifths,  thirds,  and 
fourths  can  not  be 
directly  added.  I 
therefore  reduce 
them  to  the  simi- 
lar  fractions   |J, 

2  0     o  ri  rl    4  5 

Qjj,  ana  g^. 

Then,  adding  the  numerators  48,  20,  and  45,  I  obtain  113 ;  and 
since  the  parts  are  sixtieths,  I  write  -Vif  =  l|f  ^^^  ^^^  required 
sum. 

The  common  denominator  of  the  similar  fractions  need  be  writ- 
ten but  once, — as  shown  in  the  second  process. 


-  +  - 


+     1     = 


U8-\-20-\-k5 
60 


113  _    y53 
60    ~      60 


Ex.  2.  What  is  the  sum  of  2f,  Y,  and  4f  ? 

Explanation. — Since  f  equal  f|,  and  |  equal  ^f, 
24  equal  2|f,  and  4|  equal  4^5.  Adding  the 
fractions,  I  have  f  f ,  or  If^ ;  and  I  write  the  -^^  in 
the  result. 

I  then  add  the  1  with  the  given  integers,  and  write 
the  sum,  14,  in  the  result,  making  14/^,  the  re- 
quired sum. 

Pkoblems. 


Process. 

7  =  7 


m24l3^-f4f  +  |? 
11.  8f-f  61+271? 
i^.  3^4-5  +  10^3^? 


Add  What  is  the  sum  of 

1.  ^  and  A-  4>  2f  and  4f.  7.  25^+1^ 

2.  ^  and  ^^.  5.  5-J  and  3f  8.  tV+I +  1? 
S.  1 1,  and  i.  6.  f +|+i.     9.  ^V  +  8iio  ? 

13.  ^  of  a  yard  +  fj  of  a  yard  +  -^  of  a  yard  +  ff  of  a  yard 
+  :^  of  a  yard  +  f  of  a  yard + -^^  of  a  yard  =  how  many  yards  ? 
14..  Add  60^  miles,  Y5|-  miles,  and  110^  miles. 
IS.  Add  $4|-,  $23,  $6|,  $^,  and  ISSty^r- 


168  SECOND   BOOK  IN  ARITHMETIC, 

194.  Rule  for  Addition  of  Fe actions. 

Reduce  dissimilar  to  similar  fractions,  add  the  numer- 
ators^ and  under  the  sum  write  the  common  denominator, 

a,  Beduce  fractions  to  lowest  terms,  before  reducing  to  similar  frac- 
tions. 

b.  In  aU  final  results,  reduce  fractions  to  lowest  terms,  and  improper 
fractions  to  integers  or  mixed  numbers. 

Problems. 

1,  f  of  an  acre  of  blackberries,  ^  of  an  acre  of  rasp- 
berries, and  f  of  an  acre  of  strawberries  are  how  many 
acres  of  berries  ? 

^.  A  stable  keeper  bought  two  loads  of  straw,  weighing 
1-^jy  tons,  and  |-J  of  a  ton.     How  much  straw  did  he  buy  ? 

^'  tV+I +  f  =  ^^0^^'  '"any  ?  6.  Add  ^^  ^V  hh  •'^"^  H- 

■*.  A-1-  H  +lli  =^^ow  many  ?     7.  Add  I  3%,  |i>  and  if 

S,  3f  +  5|-+64  =  how  many  ?       8.  Add  10^^,  H,  27^^'  and  U^V 

9,  A  clerk  expended  -^  of  his  salary  in  travel,  ^  of  it 
for  board,  f  for  clothing,  and  -^  for  other  purposes.  What 
pai-t  of  his  salary  did  he  expend  ? 

10.  A  lady  paid  %j\  for  sewing  silk,  S^V  for  buttons, 
$y^  for  ribbon,  $f  for  a  silver  thimble,  and  8|-  for  a  pair 
of  scissors.     How  much  did  her  purchases  amount  to  ? 

11,  How  much  wood  is  there  in  -^^  of  a  cord,  3  J 
cords,  ly^  cords,  and  -||-  of  a  cord  ? 

ArlH  i  ^^-  A'  *'  ^'  ^"^  ^-     ^^-  '^+'  ^^'  ^A,  i  and  H. 
^^^  I  i^.  ff,  ^.  and  W.         i5.   19|,  ^,  65^,  A,  and  23. 

16,  From  A  to  B  is  27-^  miles,  from  B  to  C  30  miles, 
from  C  to  D  51^  miles,  and  from  D  to  E  32yVo  miles. 
What  is  the  distance  from  A  to  E  ? 

17,  In  three  pieces  of  carpeting  that  contain  3T-|  yards, 
49|-  yards,  and  50|-  yards  are  how  many  yards  ? 


SECTION  IV. 


SUBTRACTION. 

Oral  Work.— 195.  1.  Subtract  yV  from  -^■^, 

2.  How  is  one  of  two  similar  fractions  subtracted  from 
tlie  other  ? 

3.  What  similar  fractions  are  equal  to  i  and  |  ?    "What 
is  the  difference  between  ^  and  f  ?     Between  f  and  |-  ? 


Subtract 

4.  ^  from  |. 

5.  J  from  f . 
^.  I-  from  4. 


7.  -5^*  from  |. 
6*.  4  from  I". 
9.  -J  from  ^. 


What  is  the  difference  between 


10,  Hand  A? 
ii.  5^  and  2  ? 
i^.  3f  and  f  ? 

TFAe/i  ^^e  subtrahend  is  a  mixed  number,  subtract  the  fraction  first 
How  much  is 


13.  2iand  7i? 
U,  5|and  9|? 
i5.  SyVand  4:-^} 


16,  8^  less  3-5^? 
ir.   12|-less  5|-? 


-If? 


^^.  From  9^  take  5|. 
^i.  From  243^take  15|. 


i<9.  7J- 

^^.  The  minuend  is  6^-  and  the  subtrahend  is  4f .  What 
is  the  remainder  ?     (6i  =  5  +  f .) 

^J.  Ethan  gathered  \\  of  a  bushel  of  walnuts,  and  sold 
^  of  a  bushel.     What  part  of  a  bushel  had  he  left  ? 

^Jf,.  Jennie  paid  f  of  the  yearly  cost  of  Our  Young 
People^  and  her  brother  Edgar  paid  the  balance.  What 
part  of  the  cost  did  Edgar  pay  ? 

25.  One  week  A  worked  5yV  days  and  B  Sf  days.  Which 
worked  the  longer  ?     How  much  the  longer  ? 

How  much  money  shall  I  have  left 

26.  Of  $1|,  after  paying  $-/^  for  a  slate  ? 

27.  Of  $7,  after  paying  $5^  for  a  pair  of  boots  ? 

28.  Of  $8^,  after  paying  my  grocer  $6|  ? 

29.  I  owe  $5|.    If  I  pay  $-g-,  how  much  shall  I  then  owe  ? 

30.  Anna  will  be  13  years  old  -|  of  a  year  hence.   What 
How  old  was  she  2f  years  ago  ? 

H 


is  her  age  now 


170     -        SECOND   BOOK  IN  ARITHMETIC. 


31.  A  woman  having  2|  gallons  of  boiled  cider,  used  all 
but  1-|-  gallons.     How  much  cider  did  she  use  ? 

32.  Can  I  of  a  day  be  subtracted  from  |  of  a  day  ?   Why  ? 

33.  Can  |  of  a  mile  be  subtracted  from  %\  ?    Why  ? 

Written  Work 196.  Ex.  1.  Subtract 


f  from 


1  3 


FmsT  Process. 


15 
16' 


65 

'so' 


U8  _ 

'so ' 


Explanation.— Fifths  can  not  be  di- 
rectly subtracted  from  sixteenths.    I 

therefore  reduce  the  given  fractions 

-}f  and  f  to  the  similar  fractions  |J 

and  ||. 
Then,  subtracting  the  numerator  48  from 

the  numerator  65,  1  obtain  17;  and, 

since  the  fractions  are  eightieths,  I 

write  J^  for  the  required  difference. 

Ex.  2.  What  is  the  difference  between  7|  and  4|? 


.11 
■  so 


Second  Process. 

IS        S   _65—hS__  17 


Explanation.  —  7|  equal  7^,  and  4| 


Process. 


equal  4|f .  But  f^,  the  fractional  part 
of  the  subtrahend,  is  more  than  J^, 
the  fractional  part  of  the  minuend.  I 
therefore  take  1  of  the  7,  and  unite  its 
value,  |§,  with  the  J^,  thus  changing  the 
form  of  the  minuend  to  6f^, 
Then,  subtracting  4|f  from  6f^,  I  obtain  2}§,  the  required  differ- 
ence. 

Problems. 


P  S3 

^  %6 


From 

^.  if 
2.  ^ 

9. 


Take 
h 


From 


Find  the  difference  between 


5.  A  and  3^. 

6.  AandH- 


7.  if  and  H. 

8.  Jandifi. 


62 


and  the  subtrahend  is  i^. 


Take 

4.  i      A. 

The  minuend  is 
What  is  the  remainder? 

10.  tI  of  a  mile  minus  |-||-J  of  a  mile  equals  what  part 
of  a  mile  ? 
Subtract 

11.  75^  from  99. 

12.  40^  from  103. 
IS.  g^fromlOS^;^. 


U.  7^^  from  16^0. 
IS.  45^from45f 
ifi.ff  from  194. 


17.  15f|from21-^. 

18.  SSfl  from  99yio' 

19.  235f|from532f|. 


FR  ACTIO  NS.—S  UBTRACTIO  N.  171 

197.  EuLE  FOR  Subtraction-  of  Fractions. 

Rediice  dissimilar  to  similar  fractions^  suhtract  the 
numerator  of  the  subtrahend  from  the  numerator  of  the 
m^inuend^  and  under  the  difference  write  the  common 
denominator. 

See  remarks  under  Rule  for  Addition  of  Fractions,  page  168. 

Problems. 
Find  the  remainder  in  eacli  of  these  six  problems : 


1'  m- 

2.    U- 


S.  1-1%  minus -||. 
i.  30  minus  y\ft^. 


5,  87-1^1  less  59. 

6.  543V  less  27^. 


7.  From  ^^  of  a  cord  of  wood  a  teamster  took  ||-  of  a 
cord.     How  much  wood  remained  ? 

8.  A  farmer  bought  |^  of*  a  ton  of  plaster,  and  sowed  -^^ 
of  a  ton  on  his  clover  field.    What  part  of  a  ton  had  he  left  ? 

9.  4:^^  is  how  much  greater  than  \  ? 

10.  17-5^  is  how  much  less  than  25y\  ? 

11.  What  is  the  difference  between  f ^  and  -f^  ? 

12.  If  I  have  $5 1  and  pay  out  $|^,  how  much  money 
have  I  left? 

13.  Frank  walked  IS^yg-  miles,  and  Harry  walked  as 
far  lacking  2^  miles.     How  far  did  Harry  walk  ? 

IJf..  37f  yards  of  white  flannel  shrank  If  yards  in  dye- 
ing.    How  much  did  the  cloth  then  measure  ? 

Find  the  difference  between  each  two  numbers  following : 
15.  17.  19.  21.  23. 

1Q-JL5 


172 


SECOND   BOOK  IN  ARITHMETIC. 


25,  From  a  lot  containing  ^  of  an  acre  of  land  I  sold 
■^  of  an  acre  to  one  man,  and  ^  of  an  acre  to  another. 
How  much  land  had  I  left  ? 

26,  If  f  be  taken  from  a  certain  number,  the  remainder 
is  ^,    What  is  the  number  ? 

27,  If  5  be  added  to  each  term  of  f ,  is  the  value  of  the 
fraction  increased  or  diminished  ?    How  much  ? 

28,  From  the  sum  of  f  and  3|-  subtract  tlie  difference 
between  4^  and  ^\. 

29,  What  is  the  sum  of  f  and  fj  ?  What  is  their  dif- 
ference ?    What  is  the  sum  of  their  sum  and  difference  ? 


SECTION  V. 


MULTIPLICATION. 


Oral  Work.—\9^.  A. 


How  much  is 

1.  4  times  f? 

2,  3  times  ^? 
8,  1  times  -|  ? 
^.11  times  f? 


How  much 
is 


Multiply 

5.  3^  by  2  ;  by  3. 

6.  ^hy  b ;  by  9. 

7.  4A  by  5 ;  by  9. 

8.  9J-  by  3 ;  by  6. 

13.  4  times  ^  of  a  pound  ? 
H.  8  times  -f  of  a  bushel  ? 


What  is  the  product  of 
9,  5|  multiplied  by  10? 

10,  Z^  multiplied  by  8  ? 

11,  ^  multiplied  by  6  ? 

12,  7-Jv  multiplied  by  4  ? 

i^.  5  times  7^  miles  ? 
i7.  12  times  $5f? 
18.  10  times  21  yards? 


15.  1 0  times  f  of  a  gallon  ? 
7P.  How  much  do  6  bushels  of  apples  cost,  at  $f  a  bushel  ? 
6  bushels  cost  6  times  as  much  as  1  bushel,  or  6  times  $| ; 
and  6  times  $|  are  ^^-,  or  f  3f . 

20,  How  much'  do  12  ducks  cost,  at  %^  apiece  ? 

21,  How  many  acres  of  corn  will  4  men  hoe  in  a  day, 
if  they  average  f  of  an  acre  each  ? 


FRA  CTIONS.—MULTIPLICA  TION. 


VIZ 


2^2.  In  how  many  minutes  can  I  drive  my  horse  5  miles, 
if  I  drive  at  the  rate  of  a  mile  in  7f  minutes  ? 

How  much  money  will  buy 

23.  15  barrels  of  XXX  flour,  at  $7^^  a  barrel  ? 
2]^.  12  apples,  at  f  of  a  cent  apiece  ?    * 

25.  100  clothes-pins,  at  |-  of  a  cent  apiece  ? 

26.  7  yards  of  bleached  muslin,  at  $^^  per  yard  ? 

27.  1  dozen  boxes  of  layer  raisins,  at  $lyV  P^^  t)Ox  ? 

jB.  1.  How  do  3  fourths  of  a  number  compare  with  1 
fourth  of  it  ? 

3  fourths  of  any  number  are  3  times  1  fourth  of  the  niunber. 

2.  How  do  you  find  J  of  any  number? 

3.  How  do  you  find  f  of  any  number? 


How  much  is  Multiply 

4.  i-  of  20?  9.   18  by  f. 

5.  I  of  20?  10.   11  by  ^. 

6.  I  of  25?  11.  8  by -3^. 

7.  I  of  $9?  12.   12  by  f. 

8.  -I  of  $.16?  13.   12  by  5f. 

19.  Multiply  32  pounds  by  | ; 

How  much  is 

20.  3^  of  32  yards  of  ribbon  ?     ^^. 
^i.  f  of  50  pounds  of  nails  ?       2S. 

22.  -I  of  57  feet  of  gas  pipe? 

23.  f  of  34  days' work? 


Find  the  product  of 
U.  3  multiplied  by  2^. 
15.  8  multiplied  by  4f. 
i^.  9  multiplied  by  3|. 
i7.   IIJL  times  10. 
18.  2f  times  7. 

i.  e.^  find  |  of  32  pounds. 


6f  times  10  dozen  eggs? 
V^  times  5  bushels  of  oats? 

26.  $15  multiplied  by  3|^? 

27.  $  .08  multiplied  by  5^^? 

(7,  i.  How  much  will  §  of  a  yard  of  linen  cost,  at  $  .60 
a  yard  ? 

3  eighths  of  a  yard  will  cost  3  times  as  much  as  1  eighth  of  a 
yard,  or  3  times  1  eighth  of  60  cents.  1  eighth  of  60  cents  is  7  J 
cents,  and  3  times  7^  cents  are  22^  cents. 

2.  At  $8  a  ton,  how  much  does  f  of  a  ton  cost  ? 


174 


SECOND    BOOK  IN   ARITHMETIC. 


3.  A  tailor  used  -f  of  3  yards  of  silk  serge  in  lining  a 
coat.     How  many  yards  of  serge  did  he  use  ? 

^.  At  10  cents  a  yard  for  silk  braid,  how  much  will  2 
yards  cost?  How  much  will  ^  yard  cost?  How  much 
will  2|^  yards  cost  ? 

6.  I  burned  12  thousand  feet  of  gas  in  my  store  in  the 
summer,  and  3|  times  as  much  in  the  winter.  How  many 
thousand  feet  did  I  burn  in  the  winter  ? 

6,  A  hotel  in  one  month  used  20  pounds  of  coffee,  and 
8f  times  as  much  sugar.     How  much  sugar  was  used  ? 

7,  Twe  sevenths  of  tlie  49  children  in  a  district  attend 
school.     How  many  children  attend  school? 

8,  A  and  B  bought  a  mowing  machine  for  $145,  A  pay- 
of  the  cost,  and  B  -^7.     How  much  did  each  pay  ? 


mg  A 


2>.  What  fraction  is  equal  to 


i.  I  of  9  tenths  ? 
^.  fof  9  tenths? 
How  much  is 

7.  I  of  2^  bushels?   (2i  = 

8.  I  of  If  pounds? 

9.  ^  of  6|  yards  ? 
10.  I  of  lo|  gallons? 


4of|i? 


Multiply 

6.  M  by  i 


11.  I  of  12||  rods? 

12.  I  of  27^  miles? 

13.  fof  42ifeet?   (42i  =  40  +  2i) 
H.  f  of  15f  weeks? 

a.  The  word  of  between  two  numbers,  the  first  or  both  of  which 

are  fractions,  signifies  multiplication. 
&.  A  compound  fraction  is  two  or  more  numbers  connected 

by  the  word  of,  the  first,  at  least,  of  the  numbers  being  a  fraction. 

JE.  1.  \  is  how  many  twentieths  ?  How  much  is  J  of  ^  ? 
Then,  how  much  is  |-  of  -^  ? 

2.  How  much  is  f  of  |?  |  ^.  J  of  |?  U.  |  of  |?  |  5.  |  of  |? 

6.  How  much  is  f  of  4.  ? 

2  thirds  of  4  sevenths  are  2  times  as  much  as  1  third  of  4 
sevenths.  4  sevenths  are  12  twenty-firsts ;  1  third  of  12  twen- 
ty-firsts is  4  tw^enty-firsts,  and  2  times  4  twenty-firsts  are  8 
twenty-firsts. 


FRA  CTIONS.^MULTIPLICA  TION, 


175 


Multiply 

How  much  is 

What  is  the 

product  of 

S.  i  by  -A. 

P.  |byf 

m  14  by  2i. 

ii.  2|  times  f  ? 
i^.  1|  times -5^? 
i^.  fof  2f? 
U^  A  of  If  ^ 

15,  f  multiplied  by  ^V? 

16,  ^  multiplied  by  |? 

17,  fxf?        ia  f xfxi? 

18,  Ifxf?       ^0.  IxAxf? 

-p-         f  ^^.  f  of  $2|  ?  ^5.  2-1  times  f  of  a  yard  ? 

T  J  ^^.  I  of  7f  barrels?  ^^.  l|  times  2}  yards? 

^is       I  ^^'  ^  ^^*  ^^3  ^^^^"  •  ^^-  f  X  i^j-  of  6|  pounds  ? 

I  ^^.  44-  times  f  of  a  mile  ?     ^<^.  6f  times  2  J  bushels  ? 

199.  The  general  method  of  written  work  in  multi- 
plication of  fractions  is  based  upon  the  following 

Principle.  The  product  of  ttvo  fractions  equals  the 
product  of  their  numerators  divided  hy  the  product  of 
their  denominators. 

Written  Work. — Ex.  1.  What  is  the  product  of  f  and  f  ? 

Explanation.— I  multiply  2  and  4— the 
numerators  of  the  fractions  —  together 
for  the  numerator  of  the  product;  and 
3  and  5 — the  denominators  of  the  frac- 
tions— together  for  the  denominator  of 
the  product. 

Problems. 
Find  the  product  in  each  of  the  next  twelve  problems. 


Process. 

35         3X5        IS 


3. 1  xH= 


4.  If  of  11= 
5.fiof-||= 


8.mxm= 


Ex.  2.  Find  the  product  of  |  x  f  X  |f. 

I  cancel  all  factors  common  to  both  nu- 
merators and  denominators,  before  multi- 
plying. 


mioffof|  = 
^^•Ax|xii= 

i^.|ofi|xH= 


IS 


IS.  Howmucliisfxf  Xfi? 
U.  How  much  is-^  X  i|  X  ■^^? 
15.  What  number  equals 
f  offxiofil? 


^^-  Ax  1-^X1^=: how  much? 
17.  1i  times  f  of  f  of  ■^XU= 

what  number? 
^^-  f|X^Vxiof^=:howmany? 


176  SECOND   BOOK  IN  ARITHMETIC. 

19,  What  is  the  product  of  f  of  6|  times  8  ? 
6§n:^,  and  8=:f    Hence,  I  of  6§  X  8  =  1  of  ^  X  f 

20,  Multiply  3^times^  by  16.  |  ^i.  Multiply^  by  15  times  32-^. 

^'2,  What  part  of  a  melon  is  f  of  f  of  |-  of  W  of  a  melon  ? 

23,  A  man  who  owned  ^  of  a  factory,  sold  -§j  of  his 
share.     What  part  of  the  factory  did  he  sell  ? 

^^.  Find  the  cost  of  |-J-  of  a  yard  of  silk,  at  $2|  a  yard. 

25.  I  planted  ^^  of  3  acres  to  potatoes,  and  13^^  times 
as  mnch  to  corn.     How  many  acres  did  I  plant  to  corn  ? 

26,  If  a  man  can  paint  44  rods  of  picket-fence  in  a  day, 
how  many  rods  can  he  paint  in  17:^  days  ? 

300.  Ex.  Multiply  133^^  by  8.  Processes. 


Explanation. — I  mul-  - 


tiply  t\  and  13,  sep-  ^xj^i$_.        y^_ 

arately,  by  8 ;  then,       ^^       v>?—  10  1 

adding  the  results,  I      ^^       x<5-  lu  j^, 


8 


105-h 


the  re-      ISj^j  x8=  10 5 ^j 

quired  product. 
The  work  is  commonly  written  as  shown  in  the  second  process. 

When  the  multiplicand  is  a  mixed  number  and  the  multiplier  an  inte- 
ger, multiply  the  fractional  and  integral  parts  separately,  and  then  add 
the  products. 

,-.  1^.  T  Problems. 

Multiply 

1.  -^  by  17.     3.  i|  by  13.     5,  -,^  by  1,000.     7,  34^  by  7. 

^.  14  by  12.     4'  U  by  36.     6.  -^  by  588.        S,  875f  by  53. 

9.  How  many  bnshels  of  peaches  are  there  in  75  bas- 
kets, each  basket  containing  -j^  of  a  bushel  ? 

10,  A  house  painter's  wages  are  $lf  per  day.     How 
much  do  his  wages  amount  to  in  a  month,  or  26  days  ? 

11,  How  many  days'  work  can  54  men  do  in  -5^  of  a  day  ? 

12,  How  much  will  it  cost  to  keep  my  horse  a  year,  or 
52  weeks,  at  $2^  per  week  ? 


FRACTIONS.— MUL TIPLICA  TION. 


177 


128x5   =6^0 
128x5%=725i 


301,  Ex.  What  is  the  product  Peocessps. 

of  128  multiplied  by  5f?      •  i  2 

Explanation.  —  I  multiply 
128  by  f  and  by  5  separate- 
ly; then,  adding  the  partial 
products,  I  have  725J,  the 
required  product. 

The  work  is  commonly  written  as  shown  in  the  second 
process. 

When  the  multiplicand  is  an  integer  and  tlie  multiplier  a  mixed  num- 
ber, multiply  hy  the  fractional  and  integral  parts  separately,  and  then 
add  the  products. 

Problems. 


128 
H 

86i 

eit^o 

725i 


Multiply 

1.  44  by  -^. 

2,  98  by  t|. 


S.  57  by  -^.     5.  $9  by  3|-.       1  7.  4,765  by  13^. 
Jf..  23  by  1^.     6.  $3.50  by  7|.  |  8.  13,672  by  28^. 
9.  I  bought  300  pounds  of  nails,  and  used  ^  of  them  in 
building  a  barn.     How  many  pounds  of  nails  did  I  use  ? 

10.  An  ox  weighed  1,172  pounds,  and,  when  killed,  the 
beef  weighed  W  as  much.    How  much  did  the  beef  weigh  ? 

Find  the  cost  of 


11.  81|-  pounds  of  sugar,  @  12^. 

12.  37|  bushels  of  oats,  @  44^^. 


13.  72|  barrels  of  oil,  @  $5.31. 
IJf.  67-1^  tons  of  iron,  @  $62,50. 


^O^.  Rule  for  Multiplication  of  Fractions. 

I.  B educe  mixed  ^lumbers  to  irrvproj[)er  fractions^  and 
integers  to  the  form  of  fractions. 

II.  Midtiply  all  the  numerators  together  for  the  nu- 
merator^ and  all  the  denominators  together  for  the  denom- 
inator^ of  the  product. 

Problems. 

1.  What  is  the  value  of  ^  of  -/^  of  If  of  |^  of  $34  ? 

2.  How  much  can  a  dress-maker  save  in  5^  days,  if  she 
saves  %i  a  day  ? 

H  2 


178  SECOND    BOOK  IN   ARITHMETIC. 

Find  the  cost 

3.  Of  f  of  a  yard  of  oil-cloth,  at  %%  a  yard. 

J/..  Of  I-  of  a  bushel  of  clover  seed,  at  $7tV  per  bushel. 

5.  Of  %Q^  yards  of  flannel,  at  $  .12^  per  yard. 

6.  Of  125f  pounds  of  mutton,  at  6|  cents  a  pound. 


7,  Of  a  lot  of  goods  that  cost  f  of  |  of  6  times  $18 


8.  Multiply  i  of  I  of  1^  by  A  of  |^. 

9,  A  carpenter  built  15  lengths  of  board  fence,  and  each 
length  was  f f  of  a  rod  long.     How  long  was  the  fence  ? 

10.  Suppose  a  coat  costs  f  of  $12|-,  and  a  hat  costs  §  as 
much  ;  how  much  is  paid  for  the  hat  ? 

11.  If  4  barrels  of  apples  cost  $4|,  how  much  will  6  bar- 
rels cost? 

12.  At  the  rate  of  13^  miles  per  hour,  bow  many  miles 
will  a  steamboat  run  in  2f  hours  ? 

13.  What  is  ^  of  the  sum  of  |^,  ^,  and  -^  ? 

H.  Find  the  cost  of  f  of  a  dozen  eggs,  at  %-^  per 
dozen ;  and  ^  of  a  barrel  of  flour,  at  $11  per  barrel. 

15.  From  f  of  |  take  §  of  §. 

16.  A  man  who  owned  f  of  a  ship,  sold  f  of  his  share. 
What  part  of  the  ship  did  he  then  own  ? 

17.  How  much  more  will  f  of  a  yard  of  cloth  cost,  at 
$4.50  per  yard,  than  f  of  a  yard,  at  $3.75  per  yard? 

;  18.  I  bought  10  pounds  of  sugar,  ©11^^  cents ;  and  5 
pounds  of  tea,  @  87|-  cents.     How  much  did  I  expend  ? 

19.  How  much  will  a  turkey,  weighing  8^  pounds, 
cost,  at  $^  a  pound? 

W.  From  12  pieces  of  cloth  of  40 J  yards  each,  a  mer- 
chant tailor  made  48  suits,  using  5f  yards  of  cloth  for 
each  suit.     How  much  cloth  had  he  left? 


SECTION  VL 


DIVISION, 


Oral  Work.—20S.A.  Divide 

1.  f  by  4.  6,  {%  by  2. 

2.  -^  by  3.  7.  If  by  7. 
S.  2^  by  8.             <5*.  M  by  5. 

4.  2|by  7.  P.  15i|by  5. 

5,  7f  by  11.  i^  27f  by  3. 

16.  Divide  2^^  pounds  into  11  cqnal  parts — i.  e.,  find 
^  of  2-j^  pounds. 


What  is  the  quotient  of 
It  51i^  divided  by  10? 

12,  2ll  divided  by  8  ? 

13,  30yV  divided  by  14? 
U.  22|  divided  by  6  ? 
15,  216f  divided  by  50? 


(  17,  3|-  dozen  by  6. 
Divide  ]  18,  2^  bushels  by  8. 
(  19,  7|  gallons  by  10. 


20.  $ibyl2. 

^i.  ^of  amileby5. 

22,  2i  yards  by  12. 


23.  1  ton  by  10. 
^y,.$liby4. 
'2S.  6|  feet  by  5. 


B.  1. 


1  chicken  costs  1  sixth  as  much  as  6  chickens,  or  1  sixth  of 
$2 1 J  and  1  sixth  of  |2f  is  1  sixth  of  $^^,  which  is  $|. 


^.  $1^  for  8  yards  of  muslin  is  how  much  for  1  yard  ? 

S.  If  11  persons  eat  9|-  pounds  (or  ^  pounds)  of  butter 
in  a  week,  how  much  butter  does  1  person  eat  ? 

^.  I  bought  4  pounds  of  fresh  fish  for  $f  (or  $f|). 
How  much  wa^  the  fish  per  pound  ^ 

5.  Julia  picked  |-  of  a  bushel  of  strawberries  in  4  hours. 
How  many  berries  did  she  pick  in  an  hour  ^ 

6.  $2f  for  7  bushels  of  oats  is  how  much  per  bushel  ? 

7.  $2^  per  dozen  for  photographs  is  how  much  apiece  ? 

8.  How  many  weeks  will  35f  pounds  of  butter  last  a 
family  that  uses  4  pounds  per  week  ? 

9.  lYf  yards  for  8  coats  is  how  much  for  1  coat  ? 


180 


SECOND    BOOK  IN  ARITHMETIC. 


€•  1.  4  is  4  of  what  number  ? 

4  is  1  fifth  of  5  times  4 ;  and  5  times  4  are  20. 

What  is  the  number  of  which 

5.  12  is  ^?         ^.  Hisi?         7.  S^is^?         a     His  I? 

10.  4  is  f  of  what  number  ? 

4  is  2  fifths  of  5  times  1  half  of  4 ;  1  half  of  4  is  2,  and  5  times 
2  are  10. 

What  is  the  number  of  which 

11.  6  is  I?   I    13.  46  is  I?   I    15.  3|is|?         17.  22is|? 

12.  9  is  I?  I    U.  24  is  |?  |    16.  ^  is  -^?       18.  35  is  |? 

iP.  $2  for  \  of  a  barrel  of  flour  is  how  much  for  1  baiTel  ? 
20.  $  .12^  for  ^  quire  of  paper  is  how  much  for  a  quire  ? 
^i.  After  paying  $f  for  railroad  fare,  I  had  ^  of  my 
money  left.     How  much  money  had  I  at  first  ? 

22.  7|  yards  are  ^  of  the  distance  across  a  bridge. 
Wliat  is  the  length  of  the  bridge  ? 

23.  $  .16  for  ^  of  a  pound  of  figs  is  how  much  for  1  pound  ? 
2Jf..  $16  for  1^  of  a  ton  of  hay  is  how  much  for  1  ton  ? 

25.  6  days  are  f  of  how  many  days  ? 

26.  9  bushels  of  plums  are  |  of  how  many  bushels  ? 

2>.  1.  What  is  the  quotient  of  5  divided  by  f  ? 

5  is  15  thirds;  1  third  is  contained  in  15  thirds  15  times» 
and  2  thirds  are  contained  in  15  thirds  1  half  of  15  times,  -which 
is  7  J  times. 

Divide  What  is  the  quotient  of         Divide 

2.  4  by  f.        6.  3  divided  by  |?  10.  Q  by  If. 

8.  2  by  |.        7.  10  divided  by  |?  11.  lb  by  3f. 

4.  5  by  |.        8.  4  divided  by  |?  i^.  10  by  2f. 

5.  5  by  J.        9.  8  divided  by  2|  (or  f)  ?       i^.  9  by  l^. 
To  what  must  12  be  reduced,  before  it  can  be  divided 

U.  By  I?  I  15.  By  I?  I  16.  By  ^^  |  ^7.  By  A?  |  18.  By  A? 


FRA  CTIONS.— DIVISION. 


181 


19,  To  what  must  any  integer  be  reduced,  before  it  can 
be  divided  by  a  fraction  ? 

W,  At  $-^  apiece,  how  many  hats  will  $2  buy  ? 

^1.  At  $^  a  bushel,  how  many  bushels  of  lime  will  $5  buy  ? 

^^.  How  many  yards  of  cambric  can  I  buy  for  $4,  at 
$f  a  yard  ? 

^3.  In  how  many  days  will  a  horse  eat  9  bushels  of 
oats,  if  he  eats  f  of  a  bushel  daily  ? 

^4^.  How  many  scarfs,  each  2|-  yards  long,  can  be  made 
from  18  yards  of  lace  ? 

^5,  How  many  hours  will  it  take  you  to  walk  20  miles, 
if  you  walk  2f  miles  an  hour  ? 

^6.  In  how  many  days  can  a  man  mow  15  acres,  if  he 
mows  If  acres  per  day  ? 

JEJ,  1.  Eeduce  f  and  f  to  thirty-fifths. 
^.  Divide  14  thirty-fifths  by  15  thirty-fifths. 
To  what  fractional  unit  are  the  given  fractions  reduced, 
T    d*  'd    ]  ^'  ^^^^^^^^^  ^y  fifths? 

Divide 
8. 


Fourths  by  ninths  ? 


5,  Fourths  by  eighths? 

6,  Twelfths  by  eighths? 


^'o  by  1. 
9,  ^  by  f . 
10.  A  by  f . 


11'  Abyi 

12.  i  by  |. 

13.  fbyf 
U^  fbyf. 


19.  f  by  3i. 

20.  I-  by  If. 

21.  6|by  16f. 
^^.  2i  by  H. 


15.  5|  by  i 
i(5.  4f  by  f . 
17.  4i  by  ,;V 
1<5.  iby2i. 

23.  How  many  hours  must  a  boy  work  to  earn  $f ,  if 
he  earns  %^  an  hour  ? 

^^.  In  how  many  days  will  a  hotel  use  -|  of  a  barrel  of 
sweet-potatoes,  if  it  uses  -|-  of  a  barrel  daily  ? 

25.  How  many  pounds  of  mixed  candies  can  I  buy  for 
|-  of  a  dollar,  at  ^  of  a  dollar  per  pound  ? 

26.  A  woman  received  $7|-  for  chickens,  at  $f  apiece. 
How  many  chickens  did  she  sell  ? 


182 


SECOND    BOOK  IN  ARITHMETIC, 


27,  At  the  rate  of  1  pound  of  cheese  for  f  of  a  pound 
of  hone  J,  how  many  pounds  of  cheese  must  a  grocer  give 
in  excliange  for  4f  pounds  of  honey  ? 

28,  If  a  cook  uses  ^\  pounds  of  soda  in  a  month,  in 
what  part  of  a  month  will  she  use  ^  of  a  pound  \ 

29,  A  wood  sawyer  sawed  and  split  12^-  cords  of  wood 
in  6^  days.     How  many  cords  did  he  average  per  day  ? 

30,  A  hop  grower  picked  7|-  tons  of  hops,  from  a  yard 
which  averaged  IJ  tons  to  the  acre.  How  many  acres 
were  there  in  the  hop-yard  ? 

^04.  The  process  of  dividing  by  a  fraction  is  based 
upon  the  following 

Petnciple.  Mtiltqyhjing  a  number  hy  the  denominator 
of  a  fraction  and  dividing  the  product  hy  the  numerator  ^ 
di/vides  the  number  by  the  fraction, 

A  fraction  is  inverted,  by  interchanging  the  places  of  its 
terms.     Hence, 

To  divide  by  a  fraction  : — Invert  the  divisor,  and  proceed  as  in  multi- 
plication. 


WriUen  Work.—E 

X.  Divide  ^  by  || 

-. 

First  Process. 

Second  Process. 

T2    '    16~~  13         15~  180  ~  9 

4 

5_  _^15  _^  v**^  — ^ 
13    '    16        T^^^t^      9 

Problems. 

r^.  Hbyf. 

^.To^TftrobyT^^. 

7.  H  by  f 

Divide  \2.  ^hy  |. 

5.^hY^.   (3f=^.) 

^-iiVbyf 

U.  •Hbyi2A. 

^.  5fbyf.      (5f-3^.) 

9.Ho^y^. 

CIO,  7Aby6A. 

13.  lOibyief. 

16.  20f  by8|. 

Divide^  11,  16f  bylO^. 

U.  ^  by  4f . 

17.  40|by5i 

^12.  31  by  lOf. 

15.  13 

{¥)byVi. 

1 

8.  25  by  T^. 

19,  If  j2^  of  a  bushel  of  mortar  covers  1  square  yard  of 
wall,  how  many  square  yards  will  5^  bushels  cover  ? 


FRACTIONS.— DIVISION, 


183 


W,  I  paid  $|f  for  8|  quarts  of  strawberries.  What  was 
the  price  per  quart  ? 

21,  In  how  many  days,  of  9^  hours  each,  can  a  man  per- 
form 738-j^  hours'  work  ? 

305.  Ex.  Divide  2,786|-  by  6. 

Explanation. — Dividing  the  integer  by  6,  I 

obtain  464,  with  a  remainder  of  2f ,  or  Y-. 
Then  dividing  the  J^  by  6, 1  obtain  |^,  which 
with  the  464  makes  464JJ,  the  required  quotient. 
When  the  dividend  is  a  large  mixed  number  and  tJie  divisor  an  integer^ 
divide  the  integi^al  part  as  in  integers,  and  the  remainder  as  in  fractions. 


Pkocess. 
6)2786i 


Divide 

by  7. 


Problems. 


2,  -^  bv  5 
S,  fi  by  16. 


i.  m\  by  25. 


5.  2|-  by  8. 

6,  8f  by  37. 


10.  2,506|  by  12. 

11.  2,000f  by  20. 

12.  5,678^  by  27. 


7.  151  by  18. 

8.  41-1  by  9. 

9.  400yV  by  23. 
-^    13.  If  f  of  a  ton  of  hay  can  be  bought  for  $15,  what 

part  of  a  ton  can  be  bought  for  $1  ? 

i^.  A  butcher  packed  |-  of  a  ton  of  beef  in  8  barrels. 
How  much  did  he  put  into  each  barrel  ? 

15.  I  bought  9  cakes  of  maple  sugar  that  weighed  4| 
pounds.     How  much  did  each  cake  weigh  1 

16.  A  broom  maker  used  22|-  pounds  of  broom-corn  for 
72  brooms.    How  much  broom-corn  did  he  use  for  1  broom  ? 

17.  $27f  for  18  turkeys  is  how  much  for  1  turkey  ? 

18.  What  is  the  average  weight  of  11  men,  whose  united 
weight  is  1,758|-  pounds? 


S06.  Ex.  Divide  31  by  |.    31  ^ 


Divide 

1.  12by|. 

2.  9  by  f . 

3.  ISSby^Sj.. 


Problems. 


I  62  by  If. 

5.  18  by  10^. 

6.  80  by  21 


7.  625  by  6i 

8.  8  by  2f . 

9.  $41  by  f. 


Process. 

31         9  _  rr9__   ciQ6 
1^7—7—  ^^7 


10.  16  hours  by  7|. 

11.  28  yards  by  5f . 

12.  84  dozens  by  1^. 


184  SECOND    BOOK  IN   ARITHMETIC. 

13.  A  shoe  dealer  paid  $15  for  a  case  of  overshoes,  at 
$1  a  pair.     How  many  pairs  of  shoes  were  in  the  case  ? 
IJf.  How  many  yards  of  silk  w^ill  $24  buy,  at  $^  a  yard  ? 

15.  A  stick  of  timber  24  feet  long  can  be  cut  into  how 
many  blocks,  each  f  of  a  foot  long  ? 

16.  A  yield  of  93  bushels  of  wheat  from  4J  acres,  is 
how  many  bushels  to  the  acre  ? 

17.  A  lawyer's  clerk  wrote  36  pages  in  6f  hours.    How 
many  pages  did  he  write  in  1  hour  ? 

18.  How  many  steps  of  2|-  feet  each  will  a  soldier  take, 
in  marching  a  mile,  or  5,280  feet  ? 

19.  $423  for  the  rent  of  a  house  18|^  months,  is  how 
much  per  month  ? 

W.  $1,500  for  2,707J  feet  of  sewer  pipe  is  how  much 
a  foot  ?  ^ 

207,  EuLE  FOR  Division  of  Fractions. 

I.  Reduce  mixed  numhers  to  improper  fractions^  and 
integers  to  the  form  of  fractions. 

II.  Invert  the  divisor^  and  proceed  a^  in  multiplication. 
Problems. 


Divide 


l.^hy^. 
2.  29  by  ^. 


3.  200^  by  5. 
Jf..  360  by  12f . 


8.  3^  by  45|. 


5.      fJbvlS^^g-. 

6.  l,81lVVby^. 

9.  A  seamstress  used  f  of  a  yard  of  linen  in  making  9 
collars.     How  much  linen  did  she  use  for  each  collar  ? 

10.  7  men  harvested  22|-  acres  of  wheat  in  a  day.    How 
many  acres  was  that  for  each  man  ? 

11.  How  many  quarts  of  oysters  can  be  bought  for  $11, 
at  $f  a  quart  ? 

12.  A  teamster  draws  3f  cords  of  stone  at  15  loads. 
How  many  cords  does  he  draw  at  each  load  ? 


FRACTIONS.-DIVISION.  185 

13.  How  much  candy  will  $|  pay  for,  at  $  |  a  pound  ? 
IJf.  At  $5|  a  bushel,  liow  much  clover  seed  will  %\^  buy  ? 
75.  12  tea-spoons  weigh  -^oi  2i  pound.    How  much  does 
1  spoon  weigh  ? 

16.  If  in  1  day  a  carpenter  can  build  4f  rods  of  picket- 
fence,  in  how  many  days  can  he  build  33  rods  ? 

17.  $1^  will  pay  for  how  much  tea,  at  $|  a  pound  ? 

18.  A  mechanic  whose  wages  are  $2|^  a  day,  receives  $12 
at  the  end  of  the  week.     How  many  days  has  he  worked  ? 

19.  In  how  many  minutes  will  a  locomotive  run  22|- 
miles,  running  at  the  rate  of  -j^  of  a  mile  per  minute  ? 

W.  When  14|  quarts  of  vinegar  cost  $f f,  how  much 
does  1  quart  cost? 

21.  If  a  farm  hand  breaks  ^  of  an  acre  of  fallow  in  a 
day,  in  how  many  days  can  he  break  13|-f  acres  ? 

22.  %1^  for  12|^  pounds  of  rice  is  how  much  for  1 
pound  ? 

23.  How  many  tons  of  coal  must  2  men  load  per  hour, 
to  load  23  3?3-  tons  in  9f  hours  ? 

2^.  -^  of  2|  -^  I  of  8f  rr:  what  number  ? 

25.  Divide  |  of  3|  by  7|-  times  15. 

26.  Find  the  quotient  of  15|-  times  4|-^24|-  times  |  of  7f . 

27.  Divide  2|  times  lf|  hj^oi-^  of  2-^. 

A  complex  fraction  has  a  fraction  or  a  mixed  number  in 
one  or  both  terms. 

What  is  the  vakie  of 


Simplify  the  complex  fractions 
24|  •    18 


SO. 


1^.  Divide  the  sum  of  9|  x  ^^  and  A  of  2^  x  75  by 
their  difference. 

Note.— For  outlines  of  fractions  for  review,  see  page  272. 


SECTION  VII. 

GENERAL  REVIEW  PROBLEMS  IN  FRACTIONS. 

Oral  Work.—L  Add  18^  and  7f . 

2.  I  sold  m J  horse  for  $25  more^  than  I  paid  for  him, 
and  gained  \  of  his  cost.     For  how  much  did  I  sell  him  ? 

3,  If  -^  of  $30  is  the  cost  of  5  bushels  of  tomatoes, 
what  is  the  cost  of  6  bushels  ? 

^.  A  gentleman  paid  $362^  for  a  pair  of  horses.    How 
much  did  each  horse  cost  him  ? 

5.  144  is  -^^  of  10  times  what  number  ? 

•  6,  One  week  a  mechanic  lost  ^  of  his  time.  How 
many  days  did  he  work  ? 

7.  I  of  27  is  what  part  of  54  ? 

8.  |-  of  18  is  f  of  what  number  ? 

9.  48  is  2  times  \  of  what  number  ? 

10,  A  owns  f  of  an  iron  mine,  B  owns  f  of  it,  and  C 
the  remainder.     What  part  of  the  mine  does  C  own  ? 

11,  A  man  owning  f  of  a  boat,  sold  f  of  his  share  for 
$10.     What  was  the  value  of  the  boat,  at  the  same  rate  ? 

12,  A  fruit  grower  sold  9  bushels  of  plums,  and  had  \  of 
his  plums  left.    How  many  bushels  of  plums  did  he  raise  ? 

IS.  $15  are  f  of  the  monthly  rent  of  a  house.  How 
much  is  that  a  year  ? 

IJf..  If  2  boxes  of  lemons  cost  $5^,  how  many  boxes  can 
be  bought  for  $161? 

15.  James  hoed  a  piece  of  corn  in  5|-  days,  hoeing  f  of 
an  acre  each  day.     How  many  acres  were  in  the  piece  ? 

16.  Emma  is  f  as  old  as  her  mother,  her  mother  is  ^  as 
old  as  her  grandmother,  and  her  grandmother  is  70  years 
old.     How  old  is  Emma  ? 


REVIEW  PROBLEMS  IN  FRACTIONS.  187 

Written  Work.—l.  Change  ^  to  85tlis ;  to  153ds. 

2,  Keduce  to  mixed  numbers  ^W~?  ^w^^  and  ^f  J|^. 

3.  Reduce  ^^  |^,  and  f  to  similar  fractions. 

^.  The  sum  of  three  numbers  is  35 7^^,  and  two  of 
them  are  25  6f  and  ^.     What  is  the  third  number  ? 

5,  The  minuend  is  lOSf,  and  the  remainder  is  21f. 
What  is  the  subtrahend  ? 

6.  The  quotient  is  335,  and  the  divisor  is  23^^.    What 
is  the  dividend  ? 

7.  The  dividend  is  -^  of  If,  and  the  divisor  is  |  of 
87f .     What  is  the  quotient  ? 

8,  If  13f  dozens  of  eggs  cost  $3fJ,  how  much  will  11| 
dozen  cost  ? 

^.  How  many  pounds  of  butter,  at  18|-  cents,  will  pay 
for  15|-  pounds  of  sugar,  at  10|-  cents  ? 

10.  F  sold  f  of  his  land,  then  bought  37f  acres,  and 
then  had  112^  acres.     How  much  land  had  he  at  first  ? 

11.  Divide  I  of  i  of  42f  by  i  of  -|  of  27. 

12.  I  owned  f  of  a  fruit  farm,  and  sold  f  of  my  part 
for  $3,405.     What  was  the  whole  farm  worth  ? 

13.  If  |-  of  a  yard  of  silk  is  worth  $|-,  what  is  the  value 
of  16|^  yards  ? 

H.  A  grocer,  after  selling  -|-,  f ,  -f^^  and  ^  of  a  hogshead 
of  sugar,  had  102  pounds  left.  How  many  pounds  did 
the  hogshead  contain  at  first  ? 

15.  A  saleswoman  earns  $|-  a  day,  and  her  expenses  are 
$3|-  a  week.    How  much  money  will  she  save  in  52  weeks  ? 

16.  How  much  will  f  of  3^  tons  of  coal  cost,  at  f  of 
$8|  per  ton  ? 

17.  Reduce  if,  ^V^,  iff,  ||f |,  and  W^  to  lowest  terms. 

18.  If  4  be  subtracted  from  both  terms  of  the  fraction 
■5^,  is  its  value  increased  or  diminished  ?     How  much  ? 


188  SECOND    BOOK  IN   ARITHMETIC. 

19.  What  fraction  added  to  ^  of  10|-  times  -^^  equals  1  ? 

W.  The  product  of  four  factors  is  207f ,  and  three  of 
them  are  10^,  116|f,  and  3§.  What  is  the  fourth  factor? 
Find  the  sum,  the  difference,  and  the  product  of 

21.  f  and  f.  I  22.  -J  and  2|.  |  23.  -^  and  f  |  2J,.  b\  and  7f 

25.  What  part  of  19f  is  IS^V? 

26,  Eeduce  to  similar  fractions  17|-,  |  of  -^j  and  12. 
^7.  Bought  3  crocks  of  butter  weighing  25^,  29f ,  and 

27i  pounds.  The  empty  crocks  weigh  Sy^j  ^f ,  and  T|- 
pounds.  How  many  yards  of  muslin  will  pay  for  the  but- 
ter, at  tlie  rate  of  1^  yards  per  pound  ? 

28.  Find  the  difference  between  -^  of  18|  and  |  of  17f. 

29.  Of  a  50-acre  farm  ^  was  planted  to  corn,  f  of  the 
remainder  to  potatoes,  and  the  balance  was  sown  to  wheat. 
How  many  acres  were  there  of  each  kind  of  crop  ? 

30.  A  makes  a  boot  in  f  of  a  day,  and  B  makes  one  in 
f  of  a  day.  How  many  boots  can  the  two  men  make  in 
a  day  ? 

31.  I  paid  12  cents  a  pound  for  a  live  turkey  that  weighed 
llf  pounds,  and  the  waste  in  dressing  was  f  of  its  weight. 
How  much  a  pound  did  the  dressed  turkey  cost  me  ? 

32.  B  w^alked  3f  miles  per  hour  for  7^  hours,  and  A 
walked  4^  miles  per  hour  for  3-^  hours.  Which  walked 
the  greater  distance  ?     How  much  the  greater  ? 

33.  If  a  man  can  earn  $31.25  in  26  days,  working  10 
hours  a  day,  how  much  can  he  earn  in  19  days,  working 
12  hours  a  day?  ^^  i  of  1^ 

3 If..  Divide  the  product  of  9  times  -|  and  ^ -^  by 

their  sum.  "^  T ""  ¥ 

35.  A  merchant  paid  out  ^  of  his  money,  then  ^  of 
what  remained,  and  then  -i-  of  f  of  what  then  remained. 
What  part  of  his  money  had  he  left  ? 


CHAPTER  V. 

COMPOUND    NUMBERS. 


SECTION  I. 

MEASURES  OF  EXTENSION  AND  CAPACITY. 

S08.  A  simple  number  is  a  number  that  expresses 
units  of  the  same  kind. 

A  simple  number  may  be  either  abstract  or  concrete. 

309.  A  denom^inate  number  is  a  concrete  num- 
ber that  expresses  measure,  weight,  or  money  value. 

SIO.  A  compound  nu^riber  is  a  number  that  ex- 
presses units  of  different  kinds  or  denominations. 
a,  38,  49  apples,  25  bushels,  $363  are  simple  numbers. 
h,  VI  yards,  40  days,  271  pounds,  $362  are  denominate  numbers. 
c,  3  feet  5  inches,  7  gallons  2  quarts  1  pint,  81  pounds  6  ounces 
are  compound  numbers. 


"Which  of  the  numbers  in 
the  margin 

1.  Are  simple  numbers  ? 

2.  Are  denominate  numbers  ? 


51  yards;    36  men;   12  acres. 
125  trees;  21  dozen;  258  sheep. 
5  bushels  3  pecks;    13  miles  215 
rods. 


S.  Are  compound  numbers?    18  days  6  hours  30  minutes. 

311.  The  quantity  of  all  articles  bought  and  sold  is 
determined  by  measuring^  weighing,  or  counting  them. 

313.  Linear  or  line  ineasures  are  the  measures 
used  in  measuring  distances, — as  length,  width,  thickness. 


190  SECOND   BOOK  IN  ARITHMETIC. 

The  denominations  are  the  mile  (mi.),  the  rod  (rd.),  the  yard 
(yd.),  i\iQ  foot  (ft.),  and  the  inch  (in.). 

Table  of  Linear  Measures. 


12  in.  are  1  ft. 
8  ft.  *♦  1  yd. 

^  or  5.5  yd.,  or  |    .^   ,     , 
16i  or  16.5  ft.      f 

320  rd.,  or  ) 

1,760  yd.,  or  [•  <<  1  mi. 

6,280  ft  ) 


In  measuring  goods  sold  by  the 
yard  in  length, 

4^  or  4.5  in.  are  1  eighth. 

Vm\      "   ^  1- (quarter). 

4  qr.  '*    1  yd. 


SI 3.  A  surface  is  a  figure  that  has  length  and  width. 
The  area  of  a  figure  is  the  extent  of  its  surface. 

a,  A  surface  1  inch  long  and  1  inch  wide  is  a  square  inch; 

b,  A  surface  1  foot  long  and  1  foot  wide  is  a  square  foot; 

c,  A  surface  1  yard  long  and  1  yard  wide  is  a  square  yard;  etc. 

314.  Surface  measures  are  the  measures  used  in 
computing  the  area  of  surfaces, — such  as  land,  lumber, 
flooring,  and  plastering. 

The  denominations  are  the  square  mile  (sq.  mi.),  the  acre 
(a.),  tlie  square  rod  (sq.  rd.),  the  square  yard  (sq.  yd.), 
the  square  foot  (sq.  ft.),  and  the  square  inch  (sq.  in.). 

Table  of  Square  Measures. 


144  sq.  in.  are  1  sq.  ft. 

9  sq.  ft.  *'    1  sq.  yd. 

30i  or  30.25  sq.  yd.    ♦*    1  sq.  rd. 
160  sq.  rd.  "   1  A. 

640  A.  *♦   1  sq.  mi. 


In  surveys  of  Government  lands 
160  A.  are  1  qr.  sec,  (section). 
320  A.    "    1  half  sec. 
640  A.    *'   1  sec.  (or  1  sq.  mi.). 
36  sec.  "   1  Tp.  (township). 


215.  A  solid  or  body  is  a  portion  of  matter  or  of 
space  that  has  length,  width,  and  thickness. 

a,  A  solid  whose  surfaces  are  each  1  inch  square  is  a  cubic  inch; 

b,  A  solid  whose  surfaces  are  each  1  foot  square  is  a  cubic  foot; 

c,  A  solid  whose  surfaces  are  eachl  yard  square  is  a  ctibic  yard. 


COMPOUND   NUMBERS.— MEASURES.  191 

310,  Solid  measures  are  the  measures  used  in  com- 
puting the  solid  contents  of  bodies, — such  as  timber,  wood, 
stone,  and  earth  ; — and  the  capacity  of  bins,  boxes,  etc. 

The  denominations  are  the  cord  (cd.),  the  cubic  yard  (cu. 
yd.),  the  cubic  foot  (cu.  ft.),  and  the  cubic  inch  (cu.  in.). 

Table  of  Cubic  Measures.       f  »•  C>n  public  works,  a  cubic  yard 

of  earth  is  a  standard  load. 
b.  A  pile  of  wood  or  of  rough 
stone  8  feet  long,  4  feet  wide, 
and  4  feet  high  is  1  cord. 

317.  Liquid  measures  are  the  measures  used  in 
measuring  water,  oil, milk,  molasses, wines,  and  other  liquids. 

The  denominations  are  the  gallon  (gal.),  the  quart  (qt.),  the 
pint  (pt.),  and  the  gill  (gi.). 

318.  Dry  measures  are  the  measures  used  in  meas- 
uring grain,  seeds,  fruits,  berries,  most  kinds  of  vegetables, 
lime,  charcoal,  and  some  other  articles. 

The  denominations  are  the  bushel  (bu.),  the  peck  (pk.),  the 
quart,  and  the  pint. 


1,728  cu.  in.  are  1  cu.  ft. 
27  cu.  ft.  *'  1  cu.  yd. 
128  cu.  ft.     "    1  cd. 


Table  of  Liquid  Measures. 
4  gi.  are  1  pt. 
2  pt.    "    1  qt. 
4  qt.    "    1  gal. 


Table  of  Dry  Measures. 
2  pt.  are  1  qt. 
8  qt.    "    1  pk. 
4  pk.    *'    1  bu. 


Case  I.  Reduction  Descending. 
$J19.  Seduction  Descending  is  the   process   of 
changing  a  number  to  another  of  less  unit  value. 

Changing  rods  to  yards,  feet,  or  inches  is  Reduction  Descending. 

Oral  Work. — 1.  How  many  inches  are  9  ft.  ? 

^.  How  many  inches  are  9  ft.  6  in.  ? 

S.  Keduce  2  yards  1  foot  8  inches  to  inches. 

2  yards  are  2  times  3  feet,  or  6  feet ;  and  2  yards  1  foot  are  6 
feet  plus  1  foot,  or  7  feet.  7  feet  are  7  times  12  inches,  or  84 
inches ;  and  7  feet  8  inches  are  84  inches  plus  8  inches,  or  92 
inches.     Hence,  2  yards  1  foot  8  inches  are  92  inches. 


192  SECOND   BOOK  IN  ARITHMETIC. 


Jf.,  How  many  rods  are  3  mi.  ? 
5.  Are  3  mi.  40  rd.  ? 

Reduce 

8,  1  A.  40  sq.  rd.  to  square  rods. 

P.  12  gal.  2  qt.  to  pints. 
10,  9  bu.  2  pk.  to  half-pecks. 


6.  Change  6  yards  to  feet. 

7.  Change  6  feet  to  inches. 

11,  2  bu.  1  pk.  3  qt.  to  quarts. 

12,  2  cu.  yd.  8  cu.  ft.  to  cubic  feet. 

13,  4  yd.  2  ft.  to  inches. 

IJf,,  5  gal.  1  qt.  of  catsup  will  fill  how  many  quart  bot- 
tles ?    How  many  pint  bottles  ? 

15,  How  many  quart  boxes  will  2  bu.  2  pk.  5  qt.  of 
strawberries  fill? 

16,  A  dealer  sold  3  bu.  4  qt.  of  chestnuts  by  the  pint. 
How  many  pints  did  he  sell  ? 

Reduction  Descending  is  performed  hy  multiplication. 

Written  WorA\—220.  Ex.  Eeduce  17  yd.  1  ft.  5  in. 

to  inches. 

Explanation. —1  yard  is  3  feet,  17  Process. 

yards  are  17  times  8  feet,  and  17  yards        1  J  yci,  1  ft,  5  in, 
1  foot  are  17  times  3  feet,  plus  1  foot,  o 

or  52  feet.  —     ^ 

1  foot  is  12  inches,  52  feet  are  52  times  12        ^  ^  f^' 
inches,  and  52  feet  5  inches  are  52  times        1 2 
12  inches,  plus  5  inches,  or  629  inches.     Q  2  9  in 

Hence,  17  yd.  1  ft.  5  in.  =  629  in. 

In  Reduction  Descending,  the  true  multiplicand  is  commonly  used 
as  a  multiplier,  and  the  true  multiplier  as  a  multiplicand. 

Peoblems. 
1,  Eeduce  5  mi.  187  rd.  2  yd.  1  ft.  9  in.  to  inches. 
^.  Reduce  25  sq.  rd.  3  sq.  yd.  8  sq.  ft.  to  square  inches 

Reduce  to  units  of  the  lowest  denomination  named 


S.  63  rd.  4  yd.  2  ft. 

i,  84  sq.  rd.  4  sq.  ft. 

5.  275  bu.  3  pk.  7  qt. 

6,  5  cu.  yd.  848  cu.  in. 


7.  42  yd.  3  qr.  to  inches. 

8.  34^  sq.  mi.  to  square  rods. 

9.  4.375  sq.  mi.  to  acres. 
10.  7|  cd.  to  cubic  feet. 


(5.  124  inclies  to  yards. 
Change  .|  ^.  35  yards  to  rods. 

(7.  103  pints  to  gallons. 


COMPOUND   NUMBERS.— MEASURES.  193 

Case  II.  Reduction  Ascending. 
S^l.  Heduction  Ascending    is   the   process    of 
changing  a  number  to  another  of  greater  nnit  value. 
Changing  feet  to  yards,  rods,  or  miles  is  Reduction  Ascending. 

Oral  Work. — 1,  72  inches  are  how  many  feet  ? 
^.  72  inches  are  how  many  yards  ? 
3.  96  inches  are  how  many  yards  and  feet  ? 
^.  Eeduce  207  inches  to  yards. 
1  inch  is  ^^  of  a  foot,  and  207  inches  are  ^/  feet,  or  17  feet  3 
inches ;  1  foot  is  l  of  a  yard,  and  17  feet  are  y  yards,  or  5  yards 
2  feet ;  and  5  yards  2  feet  plus  3  inches  are  5  yards  2  feet  3  inches. 

8.  150  cubic  feet  to  cords. 

9.  1,000  rods  to  miles. 
10.  320  pints  to  bushels. 

11.  100  cubic  feet  are  how  many  cubic  yards  ? 

12.  A  well  that  is  40  feet  deep  is  how  many  rods  deep  ? 

13.  If  you  buy  1  pint  of  milk  a  day,  how  many  quarts 
will  you  buy  in  60  days  ?     How  many  gallons  ? 

Reduction  Ascending  is  performed  by  division. 

Written  Work 3^^.  Ex.  Keduce  1,532  ft.  to  units 

of  higher  denominations. 

Explanation.— I  Process. 

divide  1,532  feet  3\  1,532  ft. 

bLoftetTl  6.6)510yd.^ft.{92rd.J,yd. 

yard,  and  obtain  ^9  5 

510  yards  2  feet.  Tr  q 

I  divide  the  510  1  i  n 

yards  of  this  re-  1  l.U 

suit  by  5.5,  the  ^.^ 

number  of  yards  ^    ^  ^  ^   ^         ^  ^      -,    ,       -j   ^   /*. 

inlrod,andob-      Hence,  1,532  ft.  =  92  rd.  A  yd.  2ft. 

tain  92  rods  4 

yards.     To  this  result  I  annex  2  feet,  the  first  remainder,  and 
I  have  92  rods  4  yards  2  feet,  the  required  result, 
I 


194  SECOND    BOOK  IN  ARITHMETIC, 

Problems. 

1.  How  many  feet  and  inches  are  925  inches  ? 

2.  How  many  miles  and  rods  are  2,000  rods  ? 

3.  How  many  square  miles  are  312,000  square  rods  ? 


Reduce 


8.  3,236  A.  to  sq.  mi. 

9.  185,520  cu.  in.  to  cu.  yd. 

10,  334,976  sq.  in.  to  sq.  rd. 

11,  4,713,256  cu.  in.  to  cd. 


^  4,  297  sq.  ft.  to  sq.  yd. 

5.  398  yd.  to  rods. 

6.  17,283  qt.  to  bushels. 
^  7.  1,535  pt.  to  gallons. 

12,  3,825  quarts  of  wheat  are  how  many  bushels  ? 

13,  2,469  cubic  feet  of  wood  are  how  many  cords  ? 
H,  1,727  pints  of  milk  are  how  many  gallons  ? 

^^3.  Rules  for  Reductions  of  Compound  Numbees. 

I.  For  Reduction  Descending. 

1.  Multiply  the  ^mits  of  the  highest  denomination  given, 
hy  that  number  of  the  next  lower  denomination  which 
equals  1  of  this  higher,  and  to  the  product  add  the  units 
of  the  lower  denomination, 

2,  In  like  manner,  reduce  this  result  to  units  of  the 
next  lower  denomination;  and  so  continue  until  the  given 
number  is  reduced  to  units  of  the  required  denomination, 

II.  For  Reduction  Ascending. 

1.  Divide  the  given  numher  hy  that  number  of  the  same 
denomination  which  equals  1  of  the  next  higher ;  writing 
the  quotient  as  units  of  the  higher  denomination,  and  the 
remainder  as  units  of  the  denomination  divided. 

2.  In  like  manner,  reduce  this  quotient  to  units  of  the 
next  higher  denomination;  and  so  continue  until  the  given 
7iumber  is  reduced  to  units  of  the  required  denomination, 

3.  Write  the  last  quotient  and  the  several  remainders  in 
their  order,  for  the  required  result. 


COMPOUND   NUMBERS.-^MEASURES.  195 

Problems. 

1,  How  many  tiles,  each  1  foot  long,  will  be  required 
for  1  mi.  68  rd.  2  yd.  of  tile-drain  ? 

^.  7  bu.  3  pk.  6  qt.  of  chestnuts  are  how  many  pints  ? 

3.  How  many  rods  of  fence  will  it  take  to  inclose  a 
tract  of  land  2  mi.  45  rd.  long  and  -|-  mi.  wide  ? 

^.  How  many  hills  of  corn  are  there  in  a  10-acre  field, 
if  there  is  a  hill  on  every  square  yard  ? 

5.  How  many  acres  are  there  in  an  orchard  of  6,386 
peach-trees,  each  tree  occupying  1  square  rod  of  land  ? 

6.  One  year  a  fruit  grower  had  a  crop  of  305  quarts  of 
cherries.     How  many  bushels  had  he  ? 

7.  A  perfumer  put  up  10  gallons  of  cologne  in  bottles 
that  held  1  gill  each.     How  many  bottles  did  he  fill  \ 

8.  How  many  pint  papers  of  seed-corn  are  equal  to  7 
bu.  3  pk.  5  qt.  1  pt.  ? 

9.  Christmas  week  a  grocer  sold  365  quart  cans  of  oys- 
ters.   How  many  gallons  of  oysters  did  he  sell  ? 

10.  A  seedsman  put  up  353  pint  papers  of  marrowfat 
peas.     How  many  bushels  did  he  put  up  ? 

Case  IH.  Addition. 

Oral  Work.—fi^4:.  i.  The  sum  of  9  in.,  5  in.,  11  in., 
7  in.,  3  in.,  and  10  in.  is  how  many  inches  ?  How  many 
feet  and  inches  ?    How  many  yards,  feet,  and  inches  ? 

^,  The  sum  of  7  ft.,  12  ft.,  1  ft.,  6  ft.,  and  8  ft.  is  how 
many  rods,  yards,  and  feet  ? 

3.  Add  7  gal.  3  qt.  and  2  gal.  2  qt.  1  pt. 

4-  What  is  the  sum  of  3  pk.  5  qt.  and  2  pk.  7  qt.  ? 

5  quarts  plus  7  quarts  are  12  quarts,  or  1  peck  4  quarts ;  this 
1  peck  plus  the  3  pecks  and  2  pecks  are  6  pecks,  or  1  bushel  2 
pecks ;  and  1  bushel  2  pecks  plus  4  quarts  are  1  bu.  2  pk.  4  qt. 


196  SECOND   BOOK  IN  ARITHMETIC. 

5,  Add  2  yd.  1  ft.,  1  yd.  2  ft.,  and  5  yd.  2  ft. 

6,  What  is  the  sum  of  5  sq.  yd.  7  sq.  ft.,  2  sq.  yd.  5  sq. 
ft.,  and  4  sq.  yd.  8  sq.  ft.  ? 

7.  4  yd.  +  2  yd.  +  13  yd.  +  8  yd.  +  3  yd.  —  how  many 
rd.,  yd.,  ft.,  and  in.  \ 

8.  In  3  hours  a  butcher  drove  an  ox  1  mi.  200  rd.,  1 
mi.  160  rd.,  and  2  mi.  40  rd.    How  far  was  the  ox  driven  ? 

WriUen  Work ^^5.  Ex.  Add  4  yd.  2  ft.  3  in.,  1  yd. 

1  ft.  9  in.,  2  yd.  1  ft,  2  ft.  11  in.,  and  5  yd.  5  in. 

Explanation.— I  write  the  parts  with  Process. 

units  of  like  denominations  in  the  /     /7   ^  -fi  q  ' 

same  columns,  and  begin  at  the  right  -^  V^'  -^Z^*  *^  ^^' 

to  add.  119 

The  sum  of  the  inches  is  28,  or  2  feet  ^10 
4  inches ;  and  I  write  the  4  inches  in  2      11 

the  result.  i^  0         ^ 

The  sum  of  the  2  feet  and  the  feet  in  the 


given  numbers  is  8,  or  2  yards  2  feet ;     2  rd,  3  yd.  2  ft.  ^  in. 

and  I  write  the  2  feet  in  the  result. 
The  sum  of  the  2  yards  and  the  yards  in  the  given  numbers  is  14, 

or  2  rods  3  yards,  which  I  write  in  the  result. 
The  entire  result,  2  rd.  3  yd.  2  ft.  4  in.,  is  tb5  required  sum. 

Note.— Compare  this  process  and  explanation  with  the  process  and  ex- 
planation on  page  23. 

Sd6.  The  processes  of  addition^  subtraction^  multipli- 
cation^ and  division  of  compound  nuinbers  are  similar 
to  those  employed  in  integers.  Hence^  special  rules  for 
these  processes  are  unnecessary. 

Problems. 

I  2  3 

21  gal. 3  qt. Opt.  9  A.  96  sq.rd.  29  cu.yd.16  cu.ft.  525  cu.in. 

17    1   1  11   44  10  "     14     368 
126    0   1     8   108  9     968 

43    2   0  10   56  15       3     874 


COMPOUND   NUMBERS.-'MEASURES.  197 

(  Jf.  4  yd.  2  ft.  4  in.,  3  yd.  1  ft.  8  in.,  5  yd.  2  ft.  6  in. 
Add  \  5.  20  gal.  2  qt.,  15  gal.  1  qt.,  and  14  gal.  3  gi. 

(  6.   13  bu.  +  12  bu.  3  pk.  +  lO  bu.  2  qt.  +  36  bu.  2  pk.  2  qt. 

7.  In  three  piles  of  wood  containing  5  cd.  60  cu.  ft.,  6 
cd.  96  cii.  ft.,  and  9  cd.  68  cu.  ft.,  are  how  many  cords  ? 

8.  In  four  days  a  telegraph  company  put  up  1  mi.  14  rd. 
3  yd.,  318  rd.  5  yd.,  1  ml.  39  rd.  4  yd.,  and  1  mi.  67  rd.  of 
wire.     How  much  wire  did  they  put  up  in  the  four  days  ?^ 

9.  A  farmer  raised  751  bu.  3  pk.  of  wheat,  135  bu.  1 
pk.  6  qt.  of  rye,  640  bu.  2  pk.  of  corn,  and  514  bu.  4  qt. 
of  oats.     How  much  grain  did  he  raise  ? 

10,  A  painter  used  5  gal.  3  qt.  1  pt.  of  raw  oil,  and  3  gal. 
2  qt.  1  pt.  of  boiled  oil.     How  much  oil  did  he  use  ? 

11.  There  are  35  sq.  yd.  5  sq.  ft.  of  plastering  in  the  ceil- 
ing of  a  room,  22  sq.  yd.  2  sq.  ft.  in  each  of  the  two  side 
walls,  and  17  sq.  yd.  7  sq.  ft.  in  each  of  the  two  end  walls. 
How  much  plastering  is  there  in  the  room  ? 


Case  IV.  Subtraction. 
Oral  Work.— ^^7.  Subtract 


1.  3  in.  from  4  ft.  8  in. 

2.  1  ft.  from  4  ft.  8  in. 

3.  1  ft.  3  in,  from  4  ft.  8  in. 


i.  Y  sq.  ft.  from  1  sq.  yd. 

5,  1 5  cu.  ft.  from  3  cu.  yd. 

6.  120  sq.  rd.  from  5  A 


7.  Take  10  rd.  5  yd.  from  25  rd.  2  yd. 

5  yards  can  not  be  taken  from  2  yards.  But  25  rods  2  yards 
are  24  rods  7^  yards ;  5  yards  from  7^  yards  leave  2^  yards ;  10 
rods  from  24  rods  leave  14  rods;  and  14  rods  plus  2^  yards  (2 
yards  1  foot  6  inches)  are  14  rods  2  yards  1  foot  6  inches. 

What  is  the  difference  between 


8.  6  ft.  and  4  ft.  7  in.? 
a   10  yd.  and  2  ft.  9  in.? 

10,  4  qt.  and  2  qt.  3  gi.  ? 

11.  3  qt.  1  pt  and  1  gal.  ? 


12.  4  cu.  yd.  and  13  cu.  ft.  28  cu.in.? 

13.  12  sq.  yd.  3  sq.  ft.  and  5  sq.  ft.  ? 
H.  6  sq.  ft.  and  4  sq.  ft.  36  sq.  in.? 
15.  ibu.  and^pk.? 


198  SECOND   BOOK  IN  ARITHMETIC, 

16,  All  of  a  10-acre  lot  but  3  A.  75  sq.  rd.  is  meadow. 
How  many  acres  are  meadow  ? 

1^,  From  a  barrel  containing  30  gal.  of  turpentine,  14 
gal.  3  qt.  were  drawn.     How  many  gallons  were  left  ? 

18,  On  a  city  lot  30  ft.  6  in.  wide,  stands  a  house  19  ft. 
10  in.  wide.     How  much  wider  is  the  lot  than  the  house  ? 

Written  Work.  —  ^^8.  Ex.  What  is  the  difference 
between  8  yd.  1  ft.  11  in.  and  3  yd.  2  ft.  5  in.? 

Explanation.— I  write  the  units  of  the 

subtrahend  under  units  of  like  denomi-  process. 

nations  of  the  minuend,  and  begin  at  g  y^^  ^  fi^  ^j[  l^i, 

the  right  to  subtract.  ^9  /T 

5  inches  from  11  inches  leave  6  inches,  ; ; — 

which  I  write  in  the  result.  J^  yd,  ^2  ft,    6  in, 

2  feet  can  not  be  subtracted  from  1  foot ; 

but  the  8  yards  1  foot  of  the  minuend  are  7  yards  4  feet;  and  2 
feet  from  4  feet  leave  2  feet,  which  I  write  in  the  result. 

3  yards  from  the  remaining  7  yards  of  the  minuend  leave  4  yards, 
which  I  write  in  the  result. 

The  entire  result,  4  yards  2  feet  6  inches,  is  the  required  difference. 

Note.— Compare  this  process  and  explanation  with  the  process  and  ex- 
planation on  page  40. 

Pkoblems. 

1  1 

From         5  mi.  220  rd.  4  yd.  2  ft.  5  in.  9  sq.  rd.  4  sq.  yd. 

Subtract    2         264        3         2        8  _3 5 

S,  From  1  rd.  take  1  in.   |  J^,  From  8  bu.  3  pk.  2  qt.  take  4  bu.  7  qt. 

5.  From  90  cu.  yd.  subtract  39  cu.  yd.  18  cu.  ft.  966  cu.  in. 

^.113  cd.  of  wood  have  been  drawn  from  a  wood  yard 

containing  200  cd.    How  much  wood  remains  in  the  yard  ? 

7.  Last  year  23  mi.  19-4  rd.  2  yd.  of  gas  pipe  were  in  use 
in  a  certain  city,  and  now  25  mi.  49  rd.  1  ft.  are  in  use. 
How  much  pipe  has  been  laid  during  the  year  ? 

8,  In  a  20-gallon  can  are  5  gal.  2  qt.  1  pt.  of  kerosene. 
How  much  more  kerosene  will  the  can  hold  ? 


COMPOUND   NUMBERS.— MEASURES.  199 

Case  V.  Multiplication. 

Oral  Worlc. — S3d.  1.  7  times  9  inches  are  how  many 
inches  ?    How  many  feet  ? 

^.  7  times  12  cubic  feet  are  how  many  cubic  yards  ? 


Multiply  j  ' 


5.  20  A.  16  sq.  rd.  by  10. 

6.  1  yd.  3  qr.  by  8. 


3.  2  mi.  15  rd.  by  4. 
Jf.  5  cd.  20  cu.  ft.  by  3. 

7.  9  times  3  quarts  1  pint  are  how  many  gallons,  quarts, 

and  pints  ? 

9  times  1  pint  are  9  pints,  or  4  quarts  1  pint ;  and  9  times  3 
quarts  plus  4  quarts  1  pint  are  31  quarts  1  pint,  or  7  gallons  3 
quarts  1  pint. 

8.  How  much  is  20  times  9  rd.  5  yd.  ? 

9.  How  much  is  5  times  2  sq.  yd.  8  sq.  ft.  ? 

10.  My  horse  eats  2  bu.  2  pk.  2  qt.  of  oats  in  a  week. 
How  many  bushels  does  he  eat  in  4  weeks  ? 

11,  If  a  farm  hand  can  mow  2  A.  40  sq.  rd.  of  grass  in 
a  day,  how  many  acres  can  he  mow  m  a  week  ? 

Written  Work.—^^O.  Ex.  Multiply  5  mi.  72  rd.  8 
ft.  by  8. 


Explanation. — I  write  the 


Process 


multiplier  under  the  low-  5  mi.      72  rd.     S  ft. 

est  denomination    of  the  g 

multiplicand,  and  begin  at 


the  right  to  multiply.  Jf.  1  mi.  2  5  9  rd.  1  J^ft.  6  in. 

8  times  8  feet  are  64  feet, 

or  3  rods  14J  feet  (14  ft.  6  in.);  and  I  write  14  feet  6  inches  in 

the  result. 
8  times  72  rods,  plus  the  3  rods  of  the  first  partial  product  are  579 

rods,  or  1  mile  259  rods ;  and  I  write  259  rods  in  the  result. 
8  times  5  miles,  plus  the  1  mile  of  the  second  partial  product  are  41 

miles,  which  I  write  in  the  result. 
The  entire  result,  41  miles  259  rods  14  feet  6  inches,  is  the  required 

product. 
Note. — Compare  this  process  and  explanation  with  the  process  and  ex- 
planation on  page  57. 


200  SECOND    BOOK  JX   ARITHMETIC, 

Problems. 


Multiply 

1.  50  bu.  6  qt.  by  17. 

2.  9  mi.  6  in.  by  122. 

3.  129  cu.  in.  by  384. 


Jf.  36  A.  90  sq.  rd.  16  sq.  ft.  4  sq.  in.  by  8. 

5.  32  gal.  3  qt.  1  pt.  by  7. 

6,  3  mi.  280  rd.  2  ft.  by  219. 

7.  How  many  cords  of  wood  can  a  team  draw  at  18 
loads,  if  it  draws  1  cd.  44  cu.  ft.  at  each  load  ? 

8.  How  many  bushels  of  potatoes  will  a  family  use  in 
a  year,  at  the  rate  of  3  bu.  5  qt.  per  month  ? 

9.  How  much  land  is  there  in  38  building  lots,  each  lot 
containing  5  sq.  rd.  24  sq.  yd.  ? 

10.  Some  track  hands  with  a  hand-car  pass,  4  times  each 
day,  over  the  road  between  two  railroad  stations  5  mi.  213 
rd.  1  yd.  2  ft.  apart.     How  far  do  they  travel  in  26  days  ? 

Case  VI.  Division. 

Oral  TVork.—231.  1.  Find  1  half  of  6  mi.  40  rd. 


U.  1  third  of  16  sq.  yd. 
5.  1  ninth  of  20  cu.  yd. 


2,  1  sixth  of  2  ft.  (or  24  in.). 

5.  1  sixth  of  1 4  ft.  (or  1 2  ft.  +  2  ft.). 

6.  Divide  23  gallons  3  quarts  into  5  equal  parts. 

1  fifth  of  23  gallons  is  4  gallons,  and  3  gallons  remainder ;  this 

3  gaUons  plus  the  3  quarts  of  the  given  number  aie  15  quarts ; 
1  fifth  of  15  quarts  is  3  quarts ;  and  4  gallons  plus  3  quarts  are 

4  gallons  3  quarts. 

7.  Divide  40  sq.  yd.  6  sq.  ft.  into  3  equal  parts. 

8.  Divide  21  cu.  yd.  13  cu.  ft.  by  4.      |      P.  3  bu.  3  pk.  by  8. 

10.  If  a  stone-mason  lays  27  cu.  yd.  16  cu.  ft.  of  stone  in 

5  days,  how  much  does  he  lay  per  day  ? 

11.  A  yacht  sailed  59  mi.  20  rd.  in  7  hours.     What  was 
her  average  hourly  distance  ? 

12.  If  7  men  can  mow  22  A.  29  sq.  rd.  of  grass  in  a  day, 
how  much  can  1  man  mow  ? 

IS.  How  many  barrels,  each  holding  2  bu.  3  pk.,  will  be 
required  to  hold  11  bushels  of  apples  ? 


COMPOUND    NUMBERS.— MEASURES.  201 

Written    Work.-  p^^^^^^ 

d3S.    Ex.    Divide    76     ^v/v^          7-,-          7 
,  . .           ,  ,      ^           6)76  sg.rd.  10  so.  yd. 
m.  rd.  Id  sq.  yd.  by  6.  >' ^ ^--^ 

Exp.a™.-I  write  ^ ^  ^^.  ^^.  ^^  *<?•  y^.  ^  .?•/<!. 

the  dividend  and  divi- 
sor as  in  integers,  and  begin  at  the  left  to  divide. 

1  sixth  of  76  square  rods  is  12  square  rods,  and  4  square  rods  re- 
mainder ;  and  I  write  12  square  rods  in  the  result. 

The  4  square  rods  remainder  plus  the  15  square  yards  of  the  divi- 
dend are  136  square  yards ;  1  sixth  of  136  square  yards  is  22 
square  yards,  w^ith  4  square  yards  remainder ;  and  I  write  22 
square  yards  in  the  result. 

The  4  square  yards  remainder  are  36  square  feet,  and  1  sixth  of  86 
square  feet  is  6  square  feet,  which  I  write  in  the  result. 

The  entire  result,  12  sq.  rd.  22  sq.  yd.  6  sq.  ft.,  is  the  required  result. 

Note.— Compare  this  process  and  explanation  with  the  process  and  ex- 
planation on  page  86. 

Problems. 
1  £  3 

9)16mi.95rd.l4ft.     14)36  sq.mi.'74sq.rd.     30)93  cu.yd.l4  cu.ft.( 

Jf.  Divide  526  rd.  4  ft.  9  in.  into  12  equal  parts. 
6.  Divide  428  A.  50  sq.  rd.  4  sq.  ft.  by  20. 

6.  How  much  is  1  twenty-fifth  of  19  eu.  ft.  675  cu.  in.  ? 

7.  52  gal.  2  qt.  of  sirup  will  fill  how  many  kegs  that 

hold  8  gal.  3  qt.  each  ?      (Reduce  both  numbers  to  quarts.) 

8.  Divide  35  bu.  1  pk.  2  qt.  by  17  bu.  2  pk.  5  qt. 

9.  A  workman  laid  64  rd.  3  yd.  1  ft.  of  stone-wall  in 
26  days.     How  much  did  he  lay  per  day  ? 

10.  A  bridge  pier  containing  448  cu.  yd.  of  stone,  was 
built  in  36  days.  What  was  the  average  amount  of  stone 
laid  daily  ? 

11.  A  farmer  put  125  bu.  3  pk.  6  qt.  of  wheat  into  bags 
holding  1  bu.  3  pk.  6  qt.  each.     How  many  bags  did  he  fill? 

W.  If  7  men  can  mow  26  A.  40  sq.  rd.  of  grass  in  10 
hours,  how  much  can  1  man  mow  in.  1  hour  ? 

12 


SECTION  II. 

WEIGHTS. 

333.  Weight  is  the  measure  of  the  quantity  of  nmt- 
ter  in  a  body. 

234.  Avoirdupois  weights  are  the  weights  used 
for  all  the  ordinary  purposes  of  weighing. 

The  denominations  are  the  ton  (T.),  the  hundred-weight  (cwt.), 
the  pound  (lb.),  and  the  ounce  (oz.). 

Table  of  Avoirdupois  "Weights. 

196  lb.  of  flour    are  1  bar. 
200  lb.  of  beef,  ),,    ^^^ 
pork,  or  fisb  ) 


16  oz.  are  1  lb. 

100  lb.  *•  Icwt. 

20  cwt,  or  2,000  lb.,    ''IT. 


In  selling  coal,  coai-se  metals,  and  ^  ^a  lu  are  1  or 

ores  at  wholesale  ;  and  in  estimat-  I  4  ^  Q^g  ib  )  "1  c\rt 
ing  duties  on  foreign  goods  at  the  f  ^  ^^^  ^  ^^^  ^^^  ,.  ^  ^  * 
U.  S.  Custom-houses,  J  ^  '  ' 

Oral  Work. — 1,  Eeduce  50  lb.  8  oz.  to  ounces. 

2,  Eeduce  168  oz.  to  pounds. 

3.  Add  16  cwt.  25  lb.,  10  cwt.  70  lb.,  and  6  cwt.  50  lb. 
Jf.  Take  9  cwt.  35  lb.  from  2  T.  5  cwt. 

5.  How  much  is  6  times  2  lb.  5  oz.  ? 

6.  Divide  72  lb.  8  oz.  into  5  equal  parts. 

7.  3  lb.  8  oz.  of  nutmegs  are  how  many  ounces  ? 

8.  43  oz.  of  cheese  are  how  many  pounds  ? 

9.  75  cwt.  of  coal  are  how  many  tons  ? 

10.  5  T.  12  cwt.  of  iron  are  how  many  hundred-weight  ? 

11,  One  day  a  grocer  sold  6  lb.  8  oz.,  4  lb.  4  oz.,  10  lb. 
5  oz.,  9  lb.  7  oz.,  and  5  lb.  of  butter.  How  many  pounds 
of  butter  did  he  sell  ? 


COMPOUND   NUMBERS^  —  WEIGHTS.  203 

Problems. 

Written  Work. — 1,  4  T.  16  cwt.  83  lb.  of  hay  are  how 
many  pounds  ? 

^.  25  lb.  12  oz.  of  cinnamon  are  how  many  ounces? 
-r-r  ^         ^     \  S.  Are  6,811  pounds  of  flour? 

How  many  barrels  |  ^_  ^^^  65,015  ou.ices  of  beef? 

5  6 

Add  3  T.  6  cwt.  5  lb.  7  oz.  From  2  T.  715  lb.  3  oz. 

4      16     -     9      12  Take  1       929        1 
24        9           8      10 
15      12         46      13  I 

7         32      00  Multiply  5  T.  39  lb.  6  oz. 

9      19         00      15  by       12 

-p..   . ,     (  <9.  7  times  1,285  lb.  8  oz.  by  32. 

umae  |  ^^  ^r^  ^^^  j^^  ^^  ^^^  ^^  ^3^ 

i^.  A  manufacturer  put  up  5  T.  429  lb.  of  saleratus  in 
quarter-pound  packages.  How  many  packages  did  he  put  up  ? 

11,  How  many  tons  of  bluing  will  a  manufacturer  use 
in  putting  up  845,000  1-ounce  boxes  ? 

12,  A  freight-car  was  loaded  with  3  T.  8  cwt.  48  lb.  of 
groceries,  3  T.  19  cwt.  40  lb.  of  hardware,  1  T.  1  cwt.  94 
lb.  of  furniture,  and  18  cwt.  64  lb.  of  dry  goods.  How 
much  freight  was  in  the  car  ? 

13,  A  crock  of  butter  weighed  44  lb.  6  oz.,  and  the  crock 
weighed  7  lb.  10  oz.     How  much  did  the  butter  weigh  ? 

i^.  What  is  the  total  weight  of  45  loads  of  coal,  each 
weighing  1  T.  375  lb.  ? 

15,  A  farmer  cut  28  T.  1,375  lb.  of  hay  from  15  acres 
of  meadow.     What  was  the  yield  per  acre  ? 

16,  How  many  days  will  6  bar.  84  lb.  of  flour  last  a 
family  that  uses  3  lb.  8  oz.  per  day  ? 

17,  The  total  weight  of  13  cheeses  is  440  lb.  6  oz. 
What  is  their  average  weight? 


SECTIOIT  III. 


TIME. 


S35.  Time  is  a  limited  portion  of  duration. 

The  denominations  are  the  century^  the  year  (yr.),  the  month 
(mo.),  the  week  (wk.),  the  day  (da.),  the  hour  (h.),  the 
minute  (min.),  and  the  second  (sec). 


Table  of  Time. 


60  sec. 
60  min. 
24  11. 
7  da. 
G2  wk.  1  da. ,  or  \ 

365  da.,  S 
52  wk.  2  da.,  or 

366  da., 
100  yr. 


are  1  min. 
**   Ih. 
<'   1  da. 
"   1  wk. 


1  common  yr. 

j-   "   1  leap-yr. 
'*   1  century. 


a.  February  has  28  days  in  a  com- 
mon year,  and  29  in  a  leap-year. 

h.  Every  fourth  year  from  the  be- 
ginning of  a  century  is  a  leap-year. 

c.  In  most  business  transactions  30 
days  are  considered  a  month. 


MOXTHS  AND  DATS. 

Nos.  Names.  Days. 

1st,  January,  31 

2d,  February,  28  or  29 

3d,  March,  31 

4th,  April,  30 

6th,  May,  31 

6th,  Jnne,  30 

7th,  July,  31 

8th,  August,  31 

9th,  September,  30 

10th,  October,  31 

11th,  November,  30 

12th,  December,  31 


Oral  Work. — 1.  Change  5  min.  20  sec.  to  seconds. 
^.  Eeduce  8  wk.  4  da.  to  days.     To  hours. 
3,  Reduce  150  h.  to  days.     |     i^  Reduce  63  min.  to  hours. 

5.  Add  13  h.  17  min.,  10  h.  40  min.,  and  18  min. 

6.  What  is  the  sum  of  8  wk.  5  da.,  6  wk.  6  da.,  12  wk. 
4  da.,  and  9  wk.  3  da.  ? 

9.  Muhiply  3  wk.  5  da.  by  5. 
10.  Multiply  4  h.  13  min.  by  12. 

11.  How  much  is  1  eighth  of  49  wk.  3  da.  ? 

12.  Divide  2  da.  17  h.  20  min.  by  7. 


7.  Take  3  h.  5  min.  from  8  h.  3  min. 

8.  Take  5  da.  8  h.  from  3  wk. 


COMPOUND    NUMBERS.— TIME.  205 

13.  From  12  o'clock  noon  to  6  o'clock  40  min.  p.m.  is 
how  many  minutes  ? 

H.  Charles's  age  is  13  yr.  7  mo.,  William's  is  10  yr.  5 
mo.,  and  David's  is  12  yr.  7  mo.  What  is  the  sum  of 
their  ages? 

15.  If  the  afternoon  is  of  the  same  length  as  the  fore- 
noon, what  is  the  time  from  sunrise  to  sunset  when  the 
sun  rises  at  5  o'clock  47  minutes  ? 

16.  Edwin  is  13  yr.  7  mo.  old,  and  his  father  is  4  times 
as  old.     How  old  is  his  father  ? 

17.  If  a  railroad  train  runs  80  mi.  in  4  hours,  in  how 
many  minutes  does  it  run  1  mile  ? 

Problems. 
Written  Work. — 1.  Reduce  875,675  seconds  to  higher 
denominations. 

2.  50,400  minutes  are  how  many  weeks  ? 

3.  How  many  hours  are  there  in  leap-year  ? 

^.  Reduce  365  da.  5  h.  48  min.  49  sec.  to  seconds. 

5  6 

Add  280  da.  21  h.  From      8  wk.  3  da.  3  h.  20  min. 

150        16     48  min.     Subtract  5        0      15  9  sec. 

321        12     29 

294        15     37  7 

27)  331  da.  12  h. 

8.  Multiply  9  yr.  54  da.  37  min.  by  36. 

9.  On  the  4th  day  of  July  at  noon,  of  the  year  1876, 
how  many  minutes  of  the  year  had  passed  ? 

10.  In  how  many  days  will  a  clock  tick  1,000,000  times, 
if  it  ticks  once  every  second  ? 

11.  The  sum  of  two  numbers  is  16  yr.  28  da.,  and  one 
of  the  numbers  is  11  yr.  24  da.    What  is  their  difference  ? 


206  SECOND    BOOK  IN  ARITHMETIC. 

12,  An  express  train  runs  from  Boston  to  Albany  in 
6  h.  10  min. ;  to  Buffalo  in  9  h.  30  min.  more  ;  to  Cleveland 
in  5  li.  35  min.  more  ;  to  Indianapolis  in  8  li.  55  min.  more ; 
and  to  St.  Louis  in  9  li.  10  min.  more.  "What  is  the  run- 
ning time  of  the  train  from  Boston  to  St.  Louis  ? 

IS,  In  how  many  days  of  10  hours  each  can  a  man  chop 
25  cords  of  wood,  if  he  chops  a  cord  in  3  h.  15  min.  \ 

H,  If  a  tailor  makes  112  military  suits  in  88  wk.  4  da., 
working  10  hours  a  day,  in  what  time  does  he  make  1  suit  ? 

d36.  Ex.  How  many  years,  months,  and  days  from 
April  15,  1874,  to  July  4,  1877? 

Explanation.— I  write  the  later  Process. 

of  the  two  dates  for  the  minu-       1877  vr,  7  7no.      ^  da, 
end,  and  the  earlier  for  the  sub-       187 A  JL  16 

trahend,  writing  the  number  of 


the  year,  month,  and  day,  in  order.  3  yr,  2  mo,  1 9  da, 

I  then  subtract  as  in  other  com- 
pound numbers,  calling  30  days  a  month — as  the  number  of  days 
in  the  subtrahend  is  greater  than  that  in  the  minuend. 

Problems. 

1.  Benjamin  Franklin  died  April  17, 1790,  aged  84  yr. 
3  mo.     What  was  the  date  of  his  birth  ? 

2.  George  Washington  was  bom  Feb.  22,  1732,  and 
died  Dec.  14,  1799.     How  old  was  he  when  he  died  ? 

3.  A  note  dated  Jan.  7, 1880,  was  paid  ^ov.  4,  1881. 
How  long  did  it  remain  unpaid  ? 

Jf.  A  note  was  given  April  25, 1882,  payable  Sept.  19, 
following.     How  long  had  it  to  run  ? 

5.  What  is  the  age  to-day  of  a  person  who  was  bom 
Oct.  9,  1858  ? 

6.  Find  the  difference  in  time  between  July  2, 1881, 
and  Aug.  16,  1885. 


SECTION  IV. 

ENUMERATION  OR  COUNTING. 

337,  Many  articles  are  sold  bj  count. 
The  denominations  dozen  (doz.),  gross  (gro.),  great  gross 
(grt.  gro.),  sheet,  quire,  ream  (rm.),  and  score  are  in  com- 
mon use. 

Table  of  Counting. 


12  things  are  1  doz. 
12  doz.  *'  1  gro. 
12  gro.        "    1  grt.  gro. 


24  sheets  of  paper  are  1  qnire. 
20  quires  "    1  rm. 

20  things  of  a  kind   *'    1  score. 


Oral  Work. — 1,  800  eggs  are  how  many  dozen  ? 
^.  What  is  the  sum  of  48  doz.,  28  doz.  6,  and  5  doz.  10  ? 
3,  How  many  great  gross  are  8  times  9  gro.  6  doz.  ? 
Jf,,  Divide  32  reams  15  sheets  into  9  equal  parts. 

5.  How  much  is  \  of  32  gro.  6  doz.  ? 

6,  A  bookseller  having  10  rm.l2  quires  of  letter-paper, 
sold  16  quires  12  sheets.     How  much  paper  had  he  left  ? 

Problems. 
Written  Work. — Eeduce 


3.  3,872  pens  to  grt.  gro. 

4,  5,319  sheets  to  rm. 


-?.  3  grt.  gro.  5  gro.  6  doz.  to  dozens. 
2,  7  rm.  15  quires  19  sheets  to  sheets. 

5  i 

From       50  rm.  3  quires  5  sheets             Multiply  3  gro.  9  doz.  4 
Subtract  24         9  13  by      48 

7,  In  a  ream  of  foolscap  paper  are  how  many  sheets  ? 

8,  One  week  a  grocer  sold  23  gro.  19  doz.  clothes-pins. 
How  many  clothes-pins  did  he  sell  ? 

9,  248  pigeons  are  how  many  dozen  pairs  of  pigeons  ? 
10.  A  school  that  uses  18  crayons  a  week,  will  use  how 

many  gross  of  crayons  in  40  weeks  ? 

Note. — For  mdUnes  of  compound  numbers  for  review,  see  page  273. 


208  SECOND   BOOK  IN  ARITHMETIC. 

General  Review  Problems  in  Compound  Numbers. 

Oi*al  Work.—l,  How  much  must  I  pay  for  7  gal.  2 
qt.  of  maple  syrup,  at  20  cents  a  quart  ? 

^.  3^  a  quart  for  tomatoes  is  how  much  a  bushel  ? 

3.  A  girl  sold  10  quarts  of  blackberries  each  day  for 
18  days.     How  many  bushels  did  she  sell  ? 

4"  A  farmer  sold  to  one  customer  4  bu.  of  apples,  to 
another  5  bu.  3  pk.,  and  to  another  1  bu.  3  pk.  4  qt.  How 
many  bushels  of  apples  did  he  sell  ? 

5.  How  many  hours  and  minutes  are  there  from  5 
o'clock  15  min.  a.m.,  to  6  o'clock  45  min.  p.m.  ? 

6.  A  chain  197  feet  long  is  how  many  rods  long? 

7.  If  a  man  travels  50  rods  in  5  minutes,  in  what  time 
will  he  travel  a  mile  ? 

Find  the  cost 

8.  Of  4  quarts  of  peas,  at  $1.60  a  bushel. 

9.  Of  2  yards  of  silver  wire,  at  2  cents  an  inch. 

10.  Of  7  pounds  of  indigo,  at  2  cents  an  ounce. 

11.  Of  7  lb.  6  .oz.  of  beef,  at  16  cents  a  pound. 

12.  How  many  days  were  there  in  the  first  seven  months 
of  the  year  1880  ? 

IS.  If  a  cooper  can  make  4  barrels  in  5  hours,  in  what 
time  can  he  make  1  barrel  ? 

14^.  When  flour  is  $9  a  barrel,  how  much  will  a  4?- 
pound  sack  cost? 

15.  4,054  pound  bars  of  lead  are  how  many  tons  ? 

16.  The  f  ore  quarters  of  a  lamb  weighed  7  lb.  9  oz.  each, 
and  the  hind  quarters  8  lb.  11  oz.  each.  How  much  did 
the  lamb  weigh  ? 

17.  A  blacksmith  having  8  lb.  10  oz.  of  bar  steel,  used 
3  lb.  14  oz.     How  much  had  he  left  ? 


COMPOUND   NUMBERS.— REVIEW.  209 

18.  8  loads  of  hay,  eacli  weighing  1  T.  250  lb.,  weigh 
how  much  ? 

19.  A  man  killed  7  sheep,  and  their  united  weight  was 
280  lb.  14  oz.     What  was  their  average  weight  ? 

20.  A  man  cut  4  T.  7  cwt.  of  hay  from  3  acres.  What 
was  the  average  yield  per  acre  ? 

21.  I  sowed  20  bu.  2  pk.  of  wheat,  and  11  bu.  3  pk.  of 
barley.     How  much  more  wheat  than  barley  did  I  sow  ? 

22.  $28.80  will  pay  for  how  many  gross  of  copy-books, 
at  $  .10  apiece  ? 

23.  What  is  the  capacity  of  a  cask  that  holds  10  pail- 
ful s  of  2  gal.  3  qt.  each  ? 

^^.  If  7  men  can  mow  22  A.  20  sq.  rd.  of  grass  in  a  day, 
how  much  can  1  man  mow  ? 

25.  A  stationer,  by  selling  paper  at  a  profit  of  8  cents 
a  quire,  made  $2.40.     How  many  reams  did  he  sell  ? 

Written  Work, — 1.  In  6  successive  days  a  f iniit  can- 
ning establishment  put  up  3  bu.  1  pk.,  5  bu.  2  pk.  3  qt., 
4  bu.  3  pk.  5  qt.  1  pt.,  5  bu.  1  pk.,  6  bu.  3  qt.,  and  6  bu.  7  qt. 
of  strawberries.  How  many  bushels  of  berries  were  can- 
ned in  the  week  ? 

2.  From  a  can  holding  8  gal.  of  milk,  a  milkman  sold 
2  qt.  to  each  of  6  customers,  3  pt.  to  each  of  4  others,  and 
•J-  gal.  to  each  of  3  others.    How  much  milk  was  in  the  can  ? 

3.  If  you  can  count  75  every  minute,  how  much  time 
would  you  spend  in  counting  27,000,000  ? 

^.  In  one  lamp  I  use  2  qt.,  and  in  another  5  pt.  of  kero- 
sene weekly.    How  much  kerosene  do  I  use  in  9  weeks  ? 

5.  A  farmer  raised  488  bu.  1  pk.  of  oats  from  14  acres 
of  land.     What  was  the  yield  per  acre  ? 

6.  At  a  spice  mill  5,000  doz.  4-oz.  packages  of  spices 
are  put  up  weekly.     How  many  pounds  are  put  up  ? 


210  SECOND    BOOK  IN  ARITHMETIC. 

7,  In  a  bundle  of  latli  are  100  pieces,  eaeli  4  ft.  long. 
If  laid  end  to  end  in  a  row  upon  the  ground,  how  many 
rods  would  they  reach  ? 

8,  I  own  seven  city  lots  which  contain  42  6q.  rd.  108  sq. 
ft.,  40  sq.  rd.  194  sq.  ft.,  38  sq.  rd.  256  sq.  ft.,  36  sq.  rd.  58 
sq.  ft.,  38  sq.  rd.  110  sq.  ft.,  31  sq.  rd.  216  sq.  ft.,  and  29  sq. 
rd.  88  sq.  ft.     How  much  land  is  there  in  the  seven  lots  ? 

9,  May  10,  C  hired  a  house  at  $420  per  annum,  and 
occupied  it  until  Dec.  10.     How  much  rent  did  he  pay  ? 

10.  If  12  men  can  chop  132  cords  of  wood  in  4  days, 
how  many  cords  can  1  man  chop  in  1  day  ? 

11.  One  day  5  T.  1,875  lb.  of  mackerel  were  taken  by  a 
fishing-smack.     How  many  barrels  of  fish  were  caught  ? 

12.  If  a  grocer  uses  100  reams  2  quires  of  wrapping 
paper  in  the  308  business  days  of  a  year,  how  much  paper 
does  he  use  daily  on  an  average  ? 

13.  If  1  bu.  of  apples  will  make  3  gal.  1  qt.  of  cider,  how 
many  bushels  will  be  required  to  make  926  gal.  1  qt.  ? 

H.  In  grading  250  rods  of  a  street,  5  cu.  yd.  3  cu.  ft.  of 
gravel  were  used  to  the  rod.    How  much  gravel  was  used  ? 

15.  William  lives  93  rods  from  the  school -house.  He 
attends  school  regularly  for  a  term  of  14  weeks  of  5  days 
each,  and  goes  home  to  dinner  every  day.  How  many 
miles  does  he  travel  ? 

16.  A  farm  of  184  A.  46.25  sq.  rd.  was  divided  equally 
among  5  heirs.     How  much  land  did  each  heir  receive  ? 

17.  If  a  horse  eats  1  pk.  6  qt.  of  oats  a  day,  how  many 
days  will  5  bu.  1  pk.  last  him  ? 

18.  A  train  of  63  coal-cars  was  loaded  at  a  coal-mine  in 
Pennsylvania,  3  T.  550  lb.  of  coal  being  put  upon  each 
car.     How  much  coal  did  the  train  carry  ? 


CHAPTER   VI. 

MEASUREMENTS. 


SECTION  I. 

RECTANGLES,  TRIANGLES,  AND  TRAPEZOIDS. 

938.  Dimensions  are  length,  width,  and  thickness. 

a,  A  line  has  one  dimension, — length. 

b,  A  surface  has  two  dimensions, — length  and  width. 

c,  A  body  has  three  dimensions, — length,  width,  and  thick- 

ness. 

939.  Extension  is  that  which  has  one  or  more  of 
the  three  dimensions — length,  breadth,  thickness. 

940.  An  angle  is  the  opening  be-        ^  . 
tween  two  lines,  at  the  point  at  which         \\ 

they  meet.  ^ \  \ 

The  opening  at  the  point  c,  betw^een  the  lines  Figure  i. 

a  c  and  b  c,  Figure  1,  is  an  angle. 

a.  A  right  angle  is  one  of  the  four  equal 
angles  formed  by  the  crossing  of  two  lines. 

The  four  equal  angles  about  the  point  O, 
formed  by  the  crossing  of  two  lines,  Fig- 
ure 2,  are  right  angles. 
6,  An  acute  angle  is  less  than  a  right  angle. 
c.  An  obtuse  angle  is  greater  than  a  right  angle. 

In  Figure  1,  c  is  an  acute  angle,  and  d  is  an  obtuse  angle. 

941.  Two  lines  are  perpendicidar  to  each 
other f  when  they  form  a  right  angle  at  the  point  of 
meeting. 


212 


SECOND   BOOK  IN  ARITHMETIC. 


Figure  8. 


^43.  A  parallelogram  is  a  figure  whose  four  sides 
are  straight  lines,  and  whose  opposite  sides 
are  parallel  and  equal. 

943.  A  rectangle   is   a   parallelogram 
whose  angles  are  right  angles.  , 

S44.  A  square  is  a  rectangle  that  has 
four  equal  sides. 

a.  The  altitude  of  a  parallelogram  is  the  per- 
pendicular distance  between  its  opposite  sides. 

b.  The  diagonal  of  a  parallelogram  is  a  straight 
line  joining  opposite  angles. 

c.  The  perimeter  of  a  parallelogram  is  the        ^ 
total  length  of  its  sides. 

1.  Figures  3,  4,  5  are  parallelograms,  and  the 

lines  m  n  are  diagonals. 

2.  Figures  3  and  4  are  rectangles ;  figure  4  is 

a  square.  Figure  6. 


Figure  4. 


Case  I.  To  find  the  area  of  a  rectangle. 

Oral  Work. —  ^45.  1.  Draw  a  rectangle   8  inches 
long  and  5  inches  wide. 

^.  Divide  this  figure  into  squares,  by  drawing  lines  1 
inch  apart  from  side  to  side,  and  also  from  end  to  end. 

3,  What  are  the  dimensions  ] 
.4.  What  is  the  name 

5.  Counting  from  side  to  side,  how  many  rows  of  small 
squares  are  there  in  the  figure  ? 

6.  How  many  square  inches  are  there  in  1  row  ? 


'  V  of  each  of  the  small  squares  ? 


7.  In  2  rows? 

8,  In  3  rows  ? 


9,  In  4  rows  ? 
10.  In  5  rows? 


11.  In  6  rows? 

12.  In  7  rows? 


13,  How  many  square  inches  are  there  in  the  figure  ? 


MEASUREMENTS,— RE  CTAJSfOLES. 


213 


TT  (  H'  Are  8  times  o  squares  ? 

JbLow  many  squares  {  Z    k      .  .-        o  o 

•^     ^  [15.  Are  5  times  8  squares? 

i^.  Draw  a  square  foot,  and  divide  it  into  square  inches. 

17.  Draw  a  square  yard,  and  divide  it  into  square  feet. 

18.  Wliat  are  the  dimensions,  in  inches,  of  a  square  foot  ? 

19.  A  square  foot  is  how  many  square  inches  ? 

W.  What  are  the  dimensions,  in  inches,  of  a  square  yard  ? 

A  square  yard  is  \  ^^'  ^^'"^  ""^"^  ^^"^^'^'  .^^^^• 
^  -J  I  ^^.  How  many  square  inches  ? 

jTAe  number  of  units  in  the  area  of  a  rectangle   equals  the 
product  of  the  numbers  expressing  its  two  dimensions. 

JVrittefi  Work.  —  S46.  Ex.  What  is  the  area  of  a 
field  36  rd.  long  and  32  rd.  wide  ? 

Explanation.— Since  Process. 

the  field  is  36  rods     3  2  X  3  6  sq.  rd.  =  1,152  sq.  rd. 

long    and    32    rods 

wide,  its  area  consists  of  32  strips  of  land  of  36  square  rods 

each.     I  therefore  multiply  36  square  rods  by  32,  and  obtain 

1,152  square  rods,  the  required  area. 
a,  Numbers  expressing  width  and  length  are  frequently  written 

with  the  word  "by,"  or  the  sign  of  multiplication,  between  them. 
6.  7  by  9  inches,  or  7  x  9  inches,  means  7  inches  wide  and  9  inchea 

long. 

d47«  Dimensions  are  always  expressed  in  one  or  more 
of  the  denominations  of  linear  measure. 

"  1.  In  inches,  1  f  in  square  inches. 

a.  When  the      2.  In  feet,  in  square  feet, 

dimensions  i  3.  In  yards,    V  the  area  is  i  in  square  yards, 
are  given       4.  In  rods,  in  square  rods. 

.  5.  In  miles,    J  I  in  square  miles. 

d.  In  computations  in  measurements,  the  dimensions  given 
must  be  of  the  same  denomination ;  i.  e.,  they  mast  hava 
the  same  unit. 


7.  45.3  by  32  yd.  ? 

8.  56x18  in.? 
P.  321x94  ft.? 

10,  87.5x15.31  mi.? 


214  SECOND    BOOK  IN  ARITHMETIC. 

Problems. 
"What  is  the  area  of  a  figure 

1.  27  rods  long  and  12  rods  wide?  6.  137  by  28.5  rd.  ? 

2.  215  yards  long  and  140  yards  wide? 

3.  57.25  feet  long  and  38  feet  wide? 
Jf..  31.7  inches  long  and  9.56  inches  wide? 
5.  15  feet  3  inches  long  and  S^  inches  wide? 

11,  What  is  the  difference  in  the  areas  of  two  rectan- 
gles, one  15  rd.  long  and  18 J  ft.  wide,  and  the  other  71 
yd.  long  and  3  rd.  wide  ? 

12,  A  tinsmith  covered  a  roof  with  1,152  sheets  of  tin, 
each  sheet  covering  40  by  20  inches.  How  many  square 
feet  of  roof  w^ere  there  ? 

13,  How  many  acres  are  there  in  100  mi.  of  a  4-rd.  road  ? 

Case  II.  To  find  either  dimension  of  a  rectangle. 

^M:8.  Ex.  The  area  of  the  floor  of  a  room  is  864  sq.  ft., 
and  its  width  is  24  ft.    What  is  its  length  ? 
Process. 
86  Jf  sq.ft,  :=  86  Jfft  long  and  1ft,  wide;  and 
86^ft,-^2Jt.  =  36ft. 

Explanation. — The  area  of  a  surface  864  feet  long  and  1  foot 
wide,  is  864  square  feet;  and  the  length  of  a  floor  that  has  the 
same  area  and  is  24  feet  wide,  is  1  twenty-fourth  of  864  feet. 

I  therefore  divide  864  feet  by  24,  and  obtain  36  feet,  the  required 
length  of  the  floor. 

The  number  of  units  in  either  dimension  of  a  rectangle  equals 
the  quotient  of  the  number  expressing  the  area  divided  by  the  num- 
ber expressing  the  other  dimension. 

Note.— Read  the  problems  on  the  next  page  as  follows : 
Problem  1.  Read  as  printed.    Problem  2.  Read  72  rods  in  place  of  44  rods. 
Problem  3.  Read  2,175  square  rods  in  place  of  1,584  square  rods. 
Problem  4.  Read  the  numbers  -which  stand  at  the  right  of  the  brace, 
in  their  order,  in  places  of  the  numbers  in  problem  1. 
Read  problems  5-8,   9-12,  13-16  in  a  similar  manner. 


M£:a  s  urements.  -^tria  ngles.  215 

Problems. 
What  is  the  other  dimension  of  a  figure 

1-  i.  44    rods    long,    and    containing  )  ,^2  rd.  2  175  rd. 

1,584  square  rods?  [  *       ' 

5-8.  15    feet   wide,   and    containing)     ^  -  040^^3   u 

17,880  square  feet?  f  "^^  "'  ^'^^^^  **• 

9-12.  90.5  rods  wide,  and  containino- )  ^^^^^    ,      ^^^   . 
-.Hr.  ^  o  =•>- 156.15  rd.     320  A. 

172.4  acres?  ) 

13-16.  330   feet   lono*,  and    containing:)  -, , -,     j      ^  o^-        r.. 
0.-.0  1   o  *  V  111  yd.     2,80o  sq.  ft. 

311f  square  yards?  j  .^  '  4 

-?7.  The  area  of  a  blackboard  3|-  ft.  wide  is  37f  sq.  ft. 
What  is  its  length  ? 

18.  A  board  8  inches  wide  contains  8^  square  feet. 
What  is  its  length  ? 

19.  18f  square  yards  of  oil-cloth  cover  the  floor  of  a 
room  10|-  feet  wide.     How  many  feet  long  is  the  room? 

20.  A  contractor  received  $247.50  for  flagging  a  court- 
yard 30  feet  long,  at  $2.75  per  square  yard.  What  was 
the  width  of  the  yard  ? 

349.  A  triangle  is  a  figure  bounded  by 
three  straight  lines,  and  having  three  angles. 

a.  A  right-angled  triangle  has  one  right 

angle.     (See  Fig.  6.) 

b.  An  obtuse-angled  triangle  has  one  ob- 

tuse angle.     (See  Fig.  7.) 

c.  An  acute-angled  triangle  has  three  acute 

angles.     (See  Fig.  8.) 

d.  The  base  of  a  triangle  is  the  side  on  which 

it  is  supposed  to  stand. 

e.  The  vertex  is  the  point  opposite  to  the  base. 
/.  The  altitude  is  the  perpendicular  height  of 

the  vertex  above  the  base. 

_,  Figure  8. 

The  base  of  each  of  the  triangles,  Figures  6,  7,  8, 
is  the  side  a  b ;  the  vertex  of  each  is  the  point  c ;  and  the  altitude 
of  each  is  the  perpendicular  height  of  the  vertex  above  the  base. 


216 


SECOND   BOOK  IN   ARITHMETIC, 


Case  III.  To  find  the  area,  the  base,  or  the  altitude 
of  a  triangle. 

^50.  Ex.  1.  The  altitude  of  a  triangle  is 
37  inclies,  and  the  base  is  16  inches.  What 
is  the  area  ? 

The  diagonal  m  n  in  Figure  9  divides  the  paral- 
lelogram into  two  equal  triangles.    Hence,  Figure  9. 

The  number  of  units  in  the  Process. 

area  of  a  triangle  equals  one     16  X  37  sq.  in,  ^ 
half  the  product  of  the  num-  2  —^^o  sq.  m. 

hers  expressing  its  base  and  altitude, 

Ex.  2.  The  area  of  a  triangle  is  2  sq.  yd.  3  sq.  ft.  92  sq. 

in.,  and  its  altitude  is  1  yd.  2  in.     What  is  the  length  of 

its  base?  ^ 

Process. 

2  sq.  yd.  3  sq.  ft.  9  2  sq.  in.  =  3,116  sq.  in. ; 

and  1  yd.  2  in.  =3  8.  in. 

3 , 1 1 6  sq.  in.  =z  3 , 1 1 6  in.  long  and  1  in.  wide  ; 

3,116in.-T-38  =  82in.;  and 82 in.  x2=^164in.  =  13ft. 8 in. 

Problems. 

Given ; 
Area,  to  find 

6.  178  sq.  in. ;  base,  24  in. ;        altitude. 

7.  2,86lisq.in.;  base, 8 ft.  1  in.;  altitude. 

8.  298f  sq.  yd.;  base,  128  ft.;  altitude. 

9.  12  sq.  yd.;  altitude,  9  ft;  base. 
10.  12A.56sq.vd.;  altitude, 70|-rd.;  base. 

Case  IV.  To  find  the  area  of  a  parallelogram. 

S51.  Ex.  The  base  of  a  parallelogram  is       | 1 

123  feet,  and  its  altitude  is  96  feet.     What       \  \ 

is  its  area  ?  I  \  I  \ 

Tlie  area  of  the  parallelogram  abed  equals  the 
area  of  the  rectangle  e  f  c  d.    Hence, 


Find  the  area  of  a  triangle, 
the  base  being     and  the  altitude 

1.  42  in.,         54  in. 

2.  96  ft.,         96  ft. 

3.  18  ft.  6  in.,  24  ft.  3  in. 
Jf,  59  ft.  8  in.,  13i  yd. 
5.  141  rd.,      56  rd. 


Process. 


Figure  10. 

96  X  123  sq.ft.  =  11,808  sq.ft. 


ME  A  S  UREMENTS.-^TRAPEZOID  S.  217 

The  number  of  units  in  the  area  of  a  parallelogram  equals  the 
product  of  the  numbers  expressing  its  length  and  the  perpendicu- 
lar  distance  between  its  sides. 

Pkoblems. 
Find  the  areas  of  the  following  parallelograms : 


Base. 

Altitude. 

Base. 

Altitude. 

1. 

18  inches; 

15  inches. 

^. 

17  ft.  10  in.; 

8  ft.  4  in. 

2. 

23  feet; 

25  feet. 

5, 

5  yd.  2  ft. ; 

1  ft.  3  in. 

S, 

231  yards; 

155  yards. 

6, 

32|  ft. ; 

15.75  ft. 

Case  V.  To  find  the  area  of  a  trapezoid. 
^5^.  A  trapezoid  is  a  figure  that  has  four  sides,  two 
of  which  are  parallel. 

The  altitude  of  a  trapezoid  is  the  perpendicular  distance 

between  its  parallel  sides. 
In  Figure  11  a  b  c  d  i^  o.  trapezoid,  and  e  ^  or/^  is  its  altitude. 

Ex.  1.  The  two   parallel  sides   of  a       ^^ ^  , 

trapezoid  are  24  and  18  inches,  and  the        j  / 
altitude  is  15  inches.    What  is  the  area  ?       / 
The  area  of  the  trapezoid  abed  equals  the    ^    ^ 
area  of  the  rectangle  efg  h.    Hence,  Figure  ii. 

Process. 

18  in.  +  2  k  in.     ^  .  .  i  -t  r     rn  t       -        ore 
^ — - — =21zn.;  and  15x2 1  sq.tn.  —  3 15  sq.m. 

The  njimber  of  units  in  the  area  of  a  trapezoid  equals  the  prod- 
uct of  the  two  numbers  that  express  one  half  the  sum  of  the  two 
parallel  sides  and  the  perpendicular  distance  between  them. 

Ex.  2.  The  area  of  a  trapezoid  is  7  sq.  ft.  99  sq.  in.,  and 
the  parallel  sides  are  32  and  50  inches  long.  How  far 
apart  are  they  ? 

Process. 
7  sq.  ft.  99  sq.  in  =  1,10  7  sq.  in. ; 

82  in.^  50  in.  ^  ^^  .^,  ^^^  1^107  in.  ^1,1^27  in. 
2 

K 


218  SECOND   BOOK  IN  ARITHMETIC, 

Pkoblems. 
Find  tlie  areas  of  the  following  trapezoids : 


Parallel  sides.       Distance  apart. 

1.  10  in.,  and  27  in.;        13  in. 
40  yd.,  and  22  yd. ;      25  yd. 


Parallel  ends.     Distance  between 
Jf,  i  ft.,  and  5  rd. ;     40  rd. 
5.  13  ft.,  and  17  ft.;    9} yd. 


3.  63mi.,andl.35mi.;  300  rd.   |  G,  10  ft.,  and  2^  ft.;  14ft.8in. 

^5*1.  EuLEs  FOR  Measueements  of  Rectangles,  Tri- 
angles, AND  Trapezoids. 

I.  To  find  the  area  of  a  rectangle  :— 
Multiply  the  length  hj  the  icidth. 

II.  To  find  either  dimension  of  a  rectangle  :— 
Divide  the  area  hy  the  given  dimension. 

III.  To  find  the  area  of  a  triangle  :— 

3ftdtij[>ly  the  hase  hy  the  altitude^  and  divide  the  prod- 
uct hy  ^. 

IV.  To  find  the  area  of  a  parallelogram  :— 
Multiply  the  hase  hy  the  perpendicular  distance  he- 

tioeen  the  sides, 

V.  To  find  the  area  of  a  trapezoid  :— 

Mxdtiply  one  half  the  sum  of  the  parallel  sides  hy  the 
perpendicular  distance  hetween  tliem. 

Problems. 

/.  In  a  farm  225  rods  long  and  145  rods  wide  are  liow 
many  acres  ? 

^.  How  many  yards  of  carpeting  f  yd.  wide  will  carpet 
a  parlor  26  ft.  long  and  21|-  ft.  wide  ? 

S.  The  area  of  tlie  floor  of  a  school  room  6|^  yd.  wide  is 
83^  sq.  yd.     What  are  its  dimensions,  in  feet  and  inches  ? 

^.  The  parallel  sides  of  a  trapezoid  are  44  and  32  yards, 
and  the  distance  between  them  is  20  feet.  What  is  the 
area,  in  square  yards  % 


MEASUREMENTS.— THE   CIRCLE.  219 

6,  A  building  lot  lias  a  froi\tage  of  121  ft.,  and  its  area 
is  4,719  sq.  yd.     What  is  its  otlier  dimension  ? 

6.  The  base  of  a  triangle  is  10  ft.  10  in.,  and  its  alti- 
tude is  9  ft.  2  in.     What  is  its  area  ? 

7.  Find  the  base  of  a  triangle  whose  altitude  is  50  feet 
and  area  115  square  yards. 

8.  S  bought  33|-  acres  of  land  for  $225  per  acre,  and 
laid  it  out  into  lots  5  by  8^rods,  which  he  sold  at  $100 
each.     How  much  did  he  gain  by  the  transaction  ? 

9.  The  area  of  a  meadow  in  the  form  of  a  trapezoid  is 
3  A.  68.8  sq.  rd.,  and  its  parallel  sides  are  21  and  38|-  rd. 
What  is  its  altitude  ? 

10.  How  many  feet  of  boards  are  tliere  in  the  four  sides 
of  a  barn  30  by  40  feet,  16  feet  high  to  the  eaves,  and  24^ 
feet  high  to  the  gable  peaks  ? 


SECTIO]^^   II. 

THE  CIRCLE. 

954.  A  circle  is  a  surface  bounded  by  a  curved  line, 
every  part  of  which  is  equally  distant  from  a  point  within, 
called  the  centre. 

a.  The  circumference  of  a  circle  is  the  hne  that  bounds  it. 

b.  An  arc  of  a  circle  is  any  part  of  its  circumference. 

c.  The  diameter  of  a  circle  is  the  distance  across  it  through 

its  centre. 

d.  The  radius  of  a  circle  is  a  straight  line 

that  joins  its  centre  and  circumference. 

The  radius  of  a  circle  is  equal  to  one  half  of  its 

diameter. 
Figure  12  is  a  circle ;  a,  b,  c,  d  is  its  circumference ; 

a,  b,  c  and  d  c  are  arcs ;  o  is  its  centre ;  a  c  is  its 

diameter ;  and  a  o,  b  o^  c  o,  ot  d  o  is  a  radius. 


220  SECOND    BOOK   IN  ARITHMETIC. 

Case  I.  To  find  the  circumference  or  tlie  diameter 
of  a  circle. 

^^5.  The  circumference  of  a  circle  is  about  3|  times 
its  diameter.     Hence, 

For  ordinary  purposes, — Estimate  tlie  circumference  of 
a  circle  at  3f  times  the  diameter. 

Where  greater  accuracy  is  required — as  in  circles  more  than  100 
feet  in  diameter: — Estimate  the  circumference  at  S^^j^  times  the 

^^^'^^'•-  *  Process. 

Ex.  1.  The  diame-  g  ft,  3  in.  =  63  in. 

ter  of  a  carriage  wheel  3^  x  63  in.  =  198  in.  =  16  ft.  6  in. 

is  5  ft.  3  in.   What  is 

its  circumference ?  ^^  ^    c.^^^^^'nr-^  . 

^9  ft.  8  in.  =  356  in. 

Ex.  2.  The  cir-   3^^  i^_^  3^  =  113^\  in.=  9  ft.  5^j  in. 

cumf  erence  of  the 

top  of  a  cistern  is  29  ft.  8  in.     "What  is  the  diameter  ? 

Problems. 
Find  the  circumference 

1.  Of  a  circle  8  ft.  in  diameter.  I  <5.  Of  a  tank  11  ft.  in  diameter. 

2.  Of  a  log  32  in.  through.       j  ^.  Of  a  globe  13  in.  in  diameter. 

"5.  Of  a  circle  359  inches  in  circumference. 
Find  the  J  6.  Of  a  wheel  16  feet  6  inches  in  circumference, 
diameter     7.  Of  a  mound  534.5  yards  in  circumference. 

^8.  Of  a  clock  dial  16^  inches  in  circumference. 

Case  II.  To  find  the  area  of  a  circle. 

^56.  Any  circle  may  be  sup- 
posed to  be  divided  into  equal  tri- 
angles, the  bases  forming  the  cir- 
cumference of  the  circle,  and  the 
altitudes  being  the  radius  of  the  pj^^^.^  ^3 

circle.     Hence, 

The  number  of  units  in  the  area  of  a  circle  equals  the  product 
of  the  numbers  expressing  the  radius  and  one  half  the  circumference. 


MEASUREMENTS.— THE   CIRCLE.  221 

Ex.  What  is  the  area  of  a  circle  20  feet  in  diameter  ? 
Full  Solution. 
3^  X  20  ft.  =     62  f  ft.,  circumference, 

i  of  62^  ft.         =    3  !§■  ft.,  ^  of  circumference. 
^  of  2  0  ft.  —10  ft.,  radius. 

10  X  Sl^  sq.ft.  ~314f  sq.ft.,  area. 

PROBLEMS. 

1.  What  is  the  area  of  a  circle  113  feet  in  diameter? 

2.  Of  a  barrel  head  16  inches  in  diameter  ? 

3.  Of  a  disc  of  15  inches  radius  ? 

4^.  Of  a  lake  721  rods  in  circumference  ? 

5.  Of  a  race-course  1  mile  in  circumference  ? 

6.  Of  a  circle  1,000  yards  in  circumference  ? 

Q57.  Rules  fok  Measueements  of  Circles. 

I.  To  find  the  circumference  :— 
IfuUiply  the  diameter  ly  3^, 

II.  To  find  the  diameter:— 
Divide  the  cirGitmference  hy  3^. 

III.  To  find  the  area  :— 

Multijply  one  half  of  the  cii'cumference  hy  the  radius. 

Pkoblems. 

1.  What  is  the  difference  between  the  perimeter  of  a 
square  19|-  miles  on  each  side,  and  the  circumference  of 
the  largest  circle  that  can  be  inscribed  in  it  ? 

2.  The  perimeter  of  a  square  and  the  circumference  of  a 
circle  are  each  100  feet.    Find  the  difference  in  their  areas. 

Find  the  area 

3.  Of  the  bottom  of  a  ER.  water  tank  18  ft.  in  diameter. 
4-.  Of  a  circular  pond  93  rods  in  circumference. 

6.  Of  a  barrel  head  16  inches  in  diameter. 


SECTION  III. 


RECTANGULAR  SOLIDS. 

258.  A  rectangular   solid   is   a  body 
bounded  by  six  rectangles. 

959.  A  cube  is  a  rectangular  solid  whose 
sides  are  equal  squares. 

The  solid  contents  of  a  body,  and  the  ca- 
pacity of  a  portion  of  space  are  also  called 
cubic  contents  and  volume. 


Case  I.  To  find  the  cubic  contents 
of  a  rectangular  solid. 

If  you  place  5  rows  of  8  cubic  blocks  each, 
side  by  side,  how  many  blocks  do  you  use 

1.  For  2  of  the  rows  ? 

2.  For  3  of  the  rows? 


Figare  14. 


Figure  15. 


S.  For  4  of  the  rows  ? 

Jf.  For  the  5  rows,  or  the  layer  ? 

If  you  place  3  layers  in  a  pile,  how  many  blocks  do  you  use 

5.  For  2  of  the  layers?     |     6.  For  the  3  layers? 
How  many  cubic  inches  are  there  in  a  block  of  wood 

7,  8  inches  long,  5  inches  wide,  and  1  inch  thick  ? 

8,  8  inches  long,  5  inches  wide,  and  2  inches  thick  ? 

9,  8  inches  long,  5  inches  wide,  and  3  inches  thick  ? 

10,  What  are  the  dimensions,  in  inches,  of  a  cubic  foot  ? 
What  are  the  dimensions  of  a  cubic  yard, 

11.  Expressed  in  feet  ?       |       12.  Expressed  in  inches  ? 
How  many  cubic  inches  are  there  in  a  body  that  is  12 

inches  long,  12  inches  wide,  and 


13.  1  inch  thick  ? 
IJ^.  3  inches  thick? 


15.  5  inches  thick  ? 

16.  8  inches  thick  ? 


17.  10  inches  thick? 
IS.  12  inches  thick? 


MEASUREMENTS.— RECTANGULAR  SOLIDS.     223 


cubic  inches. 

the 

cubic  feet. 

>-  volume  - 

cubic  yards. 

is 

cubic  rods. 

,  cubic  miles. 

S60.  Ex.  Find  the  cubic  contents  of  a  block  of  wood 
18  inches  long,  13  inches  wide,  and  9  inches  thick. 
Since  the  block  is  18 
inches  long  and  13  Process, 

inches   wide,  the     9  X  13  X  18  cic.  in.— '2 ^1 06  cu.  in. 
contents  of  a  por- 
tion 1  inch  thick  are  13  times  18  cubic  inches ;  and  the  contents 
of  the  entire  block  are  9  times  13  times  18  cubic  inches.    Hence, 

The  number  of  units  in  the  cubic  contents  of  a  rectangular  solid 

equals  the  product  of  the  numbers  expressing  its  three  dimensions, 

1.  In  inches," 

When  the        2.  In  feet, 

dimensions  ■{    3.  In  yards, 

are  given         4.  In  rods, 

.  5.  In  miles,   , 
Note.— For  manner  of  reading  the  following  problems,  see  note,  page  214. 

Problems. 

What  are  the  cnbic  contents  of  a  body 

1-  5.  14  feet  long,  11  feet  wide,  and  )  (37  feet.  8.2   feet.  13 

7.5  feet  deep?  )  \      feet. 

6-10,  27  inches  long,  16  inches  wide,  )  j  5.21  inches.  4.25  inch- 

and  12  inches  high?  )  (       es.  2.9  inches. 

ll-'W.  576  ft.  by  125  ft.  by  87  ft.  ?  ^         72  yd.  61  yd.  45  yd. 

Case  II.  To  find  any  one  dimension  of  a  rectangu- 
lar solid. 

261.  Ex.  The  capacity  of  a  bin  16  feet  wide  and  7^ 
feet  deep  is  2,280  cubic  feet.     What  is  its  length  ? 

The  capacity  of  Phocess. 

a    portion    of  ^  ._  ^         vn/-v         ^ 

the  bin  1  foot  7ix  16  cu.ft.=zl20  cu.  ft. 

long  is  7i  times  2, ^80  cu.  ft.  =  2,280  ft.  by  1ft.  by  1ft. 
16  cubic  feet.  2,2 8 0  ft. -^  1 20  ^  19  ft. 

2,280  cubic  feet 

is  the  capacity  of  a  bin  2,280  feet  long,  1  foot  wide,  and  1  foot 
deep ;  and  ^-^  of  2,280  feet  is  the  length  of  a  bin  16  feet  wide 
and  7|  feet  deep.    Hence, 


224  SECOND    BOOK  IN   ARITHMETIC. 

The  number  of  units  in  any  one  of  the  three  dimensions  of  a 
rectangular  solidy  equals  the  quotient  obtained  hj  dividing  the 
number  expressing  the  cubic  contents  by  the  product  of  the  num- 
bers expressing  the  other  two  dimensions. 

Problems. 
What  is  tlie  third  dimension  of  a  body 

1-  5,  4  in.  wide  and  5  in.  liii^li,  and  )  j  5.6  in.  wide.  V^Jin.  liigli. 

containing  1,280  cu.  in.?       f  (      1  cu.  ft. 

e^lO.  12  ft.  high  and  4  ft.  thick,  and  )  j  8  ft.bigli.  6.375 ft.tliick. 

containing  6,000  cu.  ft.?       [  (      200  cu.  yd. 

11-15,  4  yd.  deep  and  12.3  yd.  wide,  )  j  3J  in.  deep.  3^-  ft.  wide. 

and  containing  190.8  cu.  ft.?  f  (      8 J  cu.  yd. 

16,  56  ft.  long  and  8  ft.  high,  and  containing  12.25  cd.? 

36^.  EuLEs  FOR  Measureme5»ts  of  Eectangular  Solids. 

I.  To  find  the  cubic  contents  :— 

MuUi])ly  the  lengthy  %i^icUh,,  and  thickness  together, 

II.  To  find  one  dimension  :— 

Divide  the  cubic  cmitents  by  the  jproduct  of  the  two 
given  dimensions. 

Problems. 

1,  How  many  loads  of  earth  must  be  removed,  in  dig- 
ging a  cellar  21  ft.  long,  18  ft.  wide,  and  6  ft.  deep  ? 

^,  How  many  cords  of  wood  are  there  in  a  pile  45  ft. 
long,  4  ft.  wide,  and  7  ft.  high  ? 

3.  How  many  half -inch  cubes  can  be  put  into  a  box,  the 
inside  measurements  of  which  are  12  by  6  by  3  inches  ? 

Jf.  If  100,000  bricks,  each  2  by  4  by  8  inches,  are  piled 
together,  how  many  cubic  yards  are  there  in  the  pile  'i 

6.  A  mow  of  hay  25.5  ft.  wide,  and  12  ft.  5  in.  high 
contains  422J-  cu.  yd.     llow  many  feet  long  is  it  ? 


SECTION  lY. 

MEASUREMENTS  APPLIED  TO  BUSINESS  AFFAIRS. 

303.  Yolume  is  measured  by  the  units  given  in  the 
following 

Table  of  Standard  Units  for  Measures  of  Volume. 
1  cubic  foot  is  1,728  cu.  in. 

1  cubic  yard  "       27  cu.  ft. 

1  bushel — even  or  stricken  measure  "  2,150.42  cu.  in. 
1  bushel — heaped  measure  **  2,688        "    " 

1  gallon — liquid  measure  "      231         "    " 

1  gallon — dry  measure  (4  peck)  "     268.8     "    " 

S64.  In  business  transactions,  quantities  and  values 
are  associated  with  certain  units  suited  to  the  various 
departments  of  business. 

In  estimating  materials  and  labor,  mechanics,  artizans, 
engineers,  and  laborers  use  the  following  units : 

I.  For  measures  of  surface. 

1.  The  square  foot^  —  for  stone  -  cutting,  flagging, 
lumber,  sawed  timber,  and  masonry. 

In  estimating  quantity  and  cost  of  matched  lumber,  .25  is  added 
to  the  surface  measure. 

2.  The  square  yard^ — for  painting,  plastering,  ceil- 
ing, paving,  bricklaying,  and  masonry. 

3.  The  square^  of  100  square  feet, — for  flooring,  roof- 
ing, and  bricklaying. 

a.  Shingles  are  estimated  to  average  4  inches  wide  across  the  butt. 

9  shingles  laid  4  inches  to  the  weather  cover  1  square  foot. 
&.  1,000  shingles  laid  4  inches  to  the  weather,  without  waste, 

cover  111  square  feet.     Allowing  for  w^aste  and  imperfections, 

1,000  shingles  are  estimated  to  cover  a  square. 
c.  In  estimating  by  the  square  yard  and  by  the  square,  the  wall 

is  supposed  to  be  12  inches  or  IJ  bricks  thick. 
K2 


226  SECOND    BOOK  IN   ARITHMETIC. 

4.  The  bunch  of  lath, — 100  pieces,  each  4  feet  long. 
A  bunch  of  lath  covers  5  square  yards  of  surface. 

II.  For  measures  of  volume. 

1.  The  ctiJbic  foot^ — for  bricklaying,  masonry,  and 
hewn  timber. 

2.  The  cubic  yard^  —  for  masonry,  excavations,  and 

embankments. 

a.  On  public  works  a  cubic  yard  of  earth  is  a  standard  load. 

b.  In  computing  the  cubic  contents  of  walls,  of  foundations,  and 
of  buildings,  the  length  of  each  wall  is  measured  on  the  outside. 
The  wall  at  each  comer  is  thus  measured  twice. 

3.  The  barrel f  of  31^  gallons ;  and 

4.  The  hogshead^  of  63  gallons, — for  the  capacity  of 
cisterns  and  reservoirs. 

u.  The  barrels  and  hogsheads  used  for  commercial  purposes  are 
not  fixed  measures, — a  barrel  containing  from  30  to  40  gallons, 
and  a  hogshead  from  60  to  125  gallons. 

h.  In  the  kerosene  trade  42  gallons  are  a  barrel. 

5.  The  perch  of  stone,— 16J  cubic  feet. 

There  is  no  standard  perch  of  masonry.  Architects  and  build- 
ers in  different  sections  of  the  country  estimate  the  perch  vari- 
ously, the  range  being  from  22  to  27  cubic  feet.  The  perch  of 
24|  cubic  feet — 16|  feet  long,  IJ  feet  thick,  and  1  foot  high — is 
more  extensively  adopted  than  any  other. 

6.  The  brick. 

Bricks  are  made  of  different  sizes,  as  follows : 


Common  brick,  8x4x2  in. 
Maine  "    7ix3|x2|" 

NorthRiver  *'     8x3ix2^** 


Philadelphia  and  )  8ix4^x2| 

Baltimore  brick,  J     *       »       s 

Milwaukee     " 


a,  27  common  bricks  piled  solid,  or  22J  laid  in  mortar,  are  esti- 
mated as  a  cubic  foot. 

b.  Bricklaying  is  also  estimated  by  tJie  tJwusand,  by  actual  count, 
or  by  measure  in  the  wall. 


BUSINESS  MEASUREMENTS.  227 

fi&5.  Allowances  or  Deductions. 

In  painting,  plastering,  ceiling,  bricklaying,  and  mason- 
ry, the  following  deductions  are  made : 

1.  Materials^ — for  openings  in  walls  (doors  and  win- 
dows). 

2.  Labor f — for  ^  the  openings,  and  ^  the  corners  (by 
special  contract  only). 

a.  Materials  are  commonly  estimated  by  solid  measures ;  and  the 
work,  by  either  solid  or  surface  measures. 

h.  In  the  lumber  trade  a  square  foot  is  called  a  hoard  foot; 
and  a  cubic  foot  is  called  a  timber  foot. 

c.  Lumber  1  inch  thick  or  less  is  sold  by  surface  measure ;  if  more 
than  1  inch  thick,  it  is  computed  at  this  thickness. 

d.  The  average  width  of  a  tapering  board  is  one  half  the  sum  of 
the  widths  of  the  two  ends ;  or,  it  is  the  width  of  the  board  at 
one  half  the  distance  between  the  ends. 

e.  In  writing  dimensions,  the  accent  mark  (')  is  often  used  instead 
of  the  abbreviation  in.     Thus,  6  in.  =  6'. 

^  /,  Per  C  or  @  C  means  per  hundred;  and  per  M  or  @  M  means 
per  thousand. 

Problems. 

1,  568.5  gallons  liqnid  measure  are  how  many  cubic 
inches  less  than  the  same  number  of  gallons  dry  measure  ? 

^.  If  8  gallons  of  water  are  in  a  tub  that  will  hold  1  bush- 
el of  wheat,  how  much  more  water  will  the  tub  hold  ? 

3.  A  tank  that  will  hold  15,000  gallons  of  oil  will  hold 
how  many  bushels  of  potatoes  ? 

J,..  A  roofer  used  840  sheets  of  16'  by  24'  tin  in  roofing 
a  house.  What  was  the  area  of  the  roof  ?  How  many 
squares  of  roofing  were  covered  ? 

6,  The  floor  of  a  public  hall  44  x  75  feet  contains  how 
many  squares  of  flooring  ?  How  many  feet  of  flooring  1^ 
inches  thick  will  be  required  for  the  floor  ? 


228  SECOND    BOOK  IN  ARITHMETIC. 

6.  How  many  elates  will  be  required  for  11.52  squares 
of  roofing,  allowing  3  slates  to  cover  a  square  foot  ? 

7.  How  many  bunches,  of  500  sliingles  each,  will  cover 
a  roof,  each  side  of  which  is  67.5  ft.  long  and  25  ft.  wide  ? 

8.  How  much  will  it  cost  to  lath  and  plaster  a  room 
36  ft.  by  24  ft.  by  12  ft.,  at  $  .30  per  square  yard  ? 

9.  A  plasterer  lathed  a  room  14  ft.  by  20  ft.,  and  11  ft. 
6  in.  high.     How  many  bunches  of  lath  did  he  use  ? 

10,  How  many  square  yards  are  there  in  the  four  walls 
and  ceiling  of  a  church  70  ft.  long,  30  ft.  wide,  and  20  ft. 
high  ? 

11.  What  will  be  the  expense  of  ceiling  the  sales  room 
of  a  store  24  ft.  by  87  ft.  6  in.,  at  $1.00  per  square  yard  ? 

12,  Tlie  grade  of  a  portion  of  street  126  ft.  long  and 
60  ft.  wide,  is  to  be  raised  1  ft.  8  in.  How  many  cubic 
yards  of  earth  will  be  required  ? 

13.  At  $.81i  a  sq.  yd.,  how  much  will  it  cost  to  pave 
a  street  J  mi.  long  and  49 J  ft.  wide  ? 

H.  A  box  of  glass  contains  50  sq.  ft.  How  many  panes 
are  there  in  a  box  of  9  x  16  glass  I 

15.  How  many  boxes  of  12  x  20  glass  will  be  required  to 
glaze  32  windows,  each  requiring  3  ft.  by  6  ft.  8  in.  of  glass? 

16.  A  stone-cutter  dressed  the  tops,  front  side,  and  two 
ends  of  three  stone  steps  8x11  in.  and  4  ft.  8  in.  long ; 
and  of  a  broad  step  of  the  same  length  and  height  and  2 
ft.  9  in.  wide.  How  much  did  he  receive  for  the  work,  at 
$  .21  per  sq.  ft.  ? 

17.  A  school-house  24  x  32  feet  on  the  ground,  the  side 
walls  16  feet  high,  and  the  gable  peaks  9  feet  higher  than 
tlie  side  walls,  was  painted  three  coats,  at  31^  cents  per 
s(|uare  yard.  Allowing  for  one  half  of  8  windows  each 
?>  X  C)^  feet,  what  was  the  cost  of  the  job  ? 


BUSINESS   MEASUREMENTS,  229 

18.  How  many  cubic  yards  of  earth  must  be  removed, 
in  digging  a  cellar  52  ft.  x  28  ft.  x  8  ft.  ? 

19.  How  many  wagon  loads  of  earth  will  be  removed, 
in  digging  a  ditch  2,000  ft.  long,  4  ft.  wide,  and  18  in. 
deep? 

W.  A  breakwater  at  the  entrance  to  a  harbor  is  1,400  ft. 
long,  13  ft.  wide,  and  20  ft.  high.  How  many  cubic  yards 
of  masonry  does  it  contain  ? 

'21.  In  digging  a  cellar  45  ft.  long  and  6  ft.  deep,  270 
loads  of  earth  were  removed.     How  wide  was  the  cellar  ? 

22.  How  many  hogsheads  of  water  will  fill  a  cistern  10 
ft.  6  in.  long,  8  ft.  3  in.  wide,  and  7  ft.  deep  ? 

23.  A  rectangular  cistern  12  ft.  long,  10  ft.  wide,  and 
9  ft.  deep  will  hold  how  many  barrels  of  water  ? 

2Ji,.  How  much  lumber  is  there  in  5  boards,  each  14  ft. 
long,  15'  wide  at  one  end,  and  8'  wide  at  the  other  ? 

25.  How  many  common  bricks  will  make  a  pile  12  feet 
long,  9  feet  wide,  and  5  feet  high  ? 

26.  The  sills  of  a  30  by  40-foot  barn  are  9'  by  13'.  How 
many  feet  of  timber  do  they  contain  ? 

27.  A  pile  of  rough  stone  47  feet  x  14  feet  x  5.5  feet 
contains  how  many  perches  ?     How  many  cords  ? 

28.  Find  the  cost  of  calcimining  the  walls  and  ceiling  of 
a  room  18  by  15  by  10^  feet,  at  15-|-  cents  per  square  yard. 

29.  In  a  stick  of  timber  12'  by  18'  and  30  ft.  long,  are 
how  many  feet  of  timber  ?     How  many  feet  of  lumber  ? 

30.  How  many  common  bricks,  laid  flatwise,  will  be  re- 
quired for  a  sidewalk  100  ft.  4  in.  long  and  4  ft.  wide  ? 

31.  At  $9.50  per  M  for  materials  and  labor,  and  allow- 
ing .025  for  openings,  how  much  must  I  pay  for  the  brick- 

^  work  in  the  walls  of  a  square-roofed  house  48  ft.  long, 
25  ft.  wide,  and  18  ft.  high,  the  walls  being  12J'  thick  ? 


230  SECOND    BOOK  IN  ARITHMETIC. 

32.  How  many  cubic  yards  of  earth  must  be  removed, 
to  grade  down  l\  A,oi  ground  1  ft.  4  in.  ? 

33.  At  $  .lOJ  per  sq.  ft.,  what  will  it  cost  to  flag  a  walk 
2  ft.  wide  surrounding  a  grass  plat  20  ft.  square  ? 

SJf,.  How  many  perches  of  stone  are  required  for  ston 
ing  a  well  30  feet  deep,  the  diameter  of  the  excavation 
being  6  ft.,  and  of  the  w^ell  hole  3  ft.  ? 

35.  In  a  stock  of  12  boards  16  ft.  long  and  10  in.  wide, 
are  how  many  feet  of  lumber  ? 

36.  A  4-inch  plank  18  ft.  long  and  16'  wide  contains 
how  many  feet  of  lumber  ? 

37.  A  stick  of  ship  timber  25  ft.  long,  1  ft.  2'  wide,  and 
10'  thick  contains  how  many  timber  feet  ? 

38.  A  stick  of  timber  48  ft.  long  and  10  in.  square  con- 
tains how  many  board  feet  ? 

39.  In  18  pieces  of  studding,  each  15  ft.  long  and  2'  by 
4',  are  how  many  feet  of  lumber  ? 

Ji.0.  A  man  used  7  cd.  of  stone  in  building  a  wall  4  ft. 
high  and  2  ft.  thick.     "What  was  the  length  of  the  wall  ? 

Jf,l.  At  $15  per  M,  how  much  must  I  pay  for  2^inch 
plank  for  a  4-foot  walk  on  the  street  sides  of  a  corner  lot 
4  rd.  by  7  rd.  12  ft.  6\  the  walk  to  be  placed  2  ft.  ^'  from 
the  fence  ? 

III.  Units  for  farm  products. 

1.  A  bushel  of  grain,  seed,  fruit,  vegetables,  and  root 
crops. 

A  cubic  foot  is  .8  of  a  bushel  €1)611  measure;  it  is  also  a  small  frac- 
tion more  than  .64  of  a  bushel  lieaped  measure.     Hence, 

To  find  the  capacity,  in  bushels,  of  any  granary,  bin, 
box,  cask,  or  other  portion  of  space. 

I.  Even  measure, — Multiply  the  contents  in  cubic  feet  by  .8. 
IT.  Heaped   measure, — Multiply  the  contents  in  cubic  feet 
by  .6^. 


BUSimJSS   3£EASURE3IENTS.  231 

a.  The  even  husliel  is  the  unit  of  measure  for  wheat  and  other 
kinds  of  grain ;  and  for  seeds  and  small  fruits, — such  as  clover 
seed,  berries,  plums,  cherries,  etc. 

h.  The  heaped  bushel  is  the  unit  of  measure  for  coal,  lime,  and 
corn  in  the  ear ;  and  for  large  fruits,  root  crops,  and  garden  veg- 
etables,—such  as  apples,  peaches,  potatoes,  turnips,  tomatoes, 
snap  beans,  green  peas,  etc. 

2.  A  ton  of  hay. 

Farmers  estimate  the  volume  of  hav  tliat  will  weigh  a 
ton,  by  the  following  units  : 

550  cu.  ft.  of  clover  or  meadow  hay  loose,  or  in  loads,  or  on 

scaffolds. 
450  cu.  ft.  of  meadow  hay  in  bays,  or  mows. 
270  cu.  ft.,  or  10  cu.  yd.,  well  settled  in  large  mows  or  stacks. 

Problems. 
There,  are  how  many  bushels 

Ji.2,  Of  wheat,  in  a  freight-car  33  ft.  long  and  S  ft.  wide, 
the  wheat  being  2  ft.  4|-'  deep  in  the  car  ? 

Jf3,  Of  potatoes,  in  a  bin  5  ft.  by  16  ft.  and  20  ft.  deep  ? 

U'  Of  coal,  in  a  wagon  box  10  ft.  6'  x  3  ft.  4^  x  1  ft.  3', 
the  box  being  even  full  ? 

JiS.  Of  corn  in  the  ear,  in  a  crib  100  ft.  long,  12  ft.  high, 
7  ft.  wide  at  the  bottom,  and  10  ft.  wide  at  the  top  ? 

Jf6.  In  three  bins— (r^^)  of  wheat,  6  ft.  by  8  ft.  by  4|-  ft. ; 
(b)  of  oats,  6  ft.  by  9J  ft.  by  4|-  ft. ;  and  {e)  of  shelled  corn, 
6  ft.  by  6  ft.  by  4^  ft.  ? 
How  many  tons  are  there 

J^7.  Of  old  hay,  in  a  mow  40  ft.  x  25  ft.  x  18  ft.  ? 

J^8,  In  a  load  of  hay  15  ft.  by  8  ft.  4^  by  7  ft.  W  ? 

Jt9.  Of  clover  hay,  on  a  scaffold  over  a  barn  floor,  30  ft. 
long,  12  ft.  wide,  and  9  ft.  high  to  the  peak  of  the  roof  ? 
Note.— For  outlines  of  measurements  for  review,  see  page  273. 


SECTION  V. 

GENERAL  REVIEW  PROBLEMS  IN  MEASUREMENTS. 

1-  Jf.  A  field  72  rods  long  and  21  rods  wide  [  43 J  rd.  long. 

contains  how  many  square  rods?  S  37.7  rd.  wide. 

5—  8,  A  carpet  7^  yards  long  and  4|-  yards  )  Gf  yd.  long. 

wide  contains  bow  many  square  yards?    [  5.625  yd.  wide. 
9-12,  A  ceiling  87.3  feet  X  21.55  feet  contains  )  34yV  ft.  by 

bow  many  square  yards?  j  25.5  ft. 

"W  hat  are  the  cubic  contents  of  a  body 
13-17,  5  yd.  long,  5  ft.  wide,  and  5'  bigb  ? )  8J-  yd. ;  8}  ft. ;  8|-'. 
18-22,  \i ft.  8'  by  5  ft.  6'  by  4  ft.  2'?        )  l\  ft. ;  4|-  ft. ;  ^^^  ft. 
23,  A  girl  covered  a  box,  4'  by  7'  by  9^',  with  gilt  paper. 
How  many  square  inches  of  paper  did  she  use  ? 

2Ji„  The  base  of  the  Great  Pyramid  of  Cheops  is  763  ft. 
square.     How  many  acres  does  it  cover  ? 

25.  What  is  the  difference  between  a  figure  that  contains 
•J-  of  a  square  foot,  and  one  that  is  ^  of  a  foot  square  ? 

26.  How  many  persons  can  be  seated  in  a  hall  120  ft. 
long  and  64  ft.  6'  wide,  allowing  5  persons  to  2  square 
yards  ? 

27.  How  much  will  it  cost  me  to  carpet  a  parlor  15  ft. 
9'  by  22  ft.  6',  with  tapestry  carpeting  f  yd.  wide,  at  $  .95 
a  yard,  running  measure  ? 

28.  At  $1.12|-  per  sq.  yd.,  how  much  must  I  pay  for  oil- 
cloth for  the  hall  of  my  house,  which  is  2f  by  8|-  yd.  ? 

29.  The  ceiling  of  a  room  is  18  feet  long,  and  its  area  is 
288  sq.  ft.     What  is  its  width  ? 

30.  What  is  the  length  of  a  carpet  for  a  flight  of  stairs 
of  18  steps,  each  10|-  in.  wide  and  7^  in.  high  ? 

31.  What  must  be  the  width  of  a  board  16  feet  long,  to 
contain  12  square  feet? 


REVIEW  PROBLEMS   IN  MEASUREMENTS,    233 

32,  "What  length  of  a  board  8  in.  wide  will  make  a  box 
cover  2  ft.  8  in.  square  ? 

33.  A  certain  rectangular  tract  of  land  32  rods  wide 
contains  120  acres.     What  is  its  length  ? 

3Ji..  If  64  yd.  of  carpeting  f  yd.  wide  will  carpet  a  room 
24  ft.  long,  what  is  the  width  of  the  room  ? 

35.  A  window  curtain  3  ft.  3  in.  wide  contains  2  sq.  yd. 
2  sq.  ft.  84  sq.  in.     "What  is  its  length  ? 

36.  How  many  pickets  3  in.  wide,  and  placed  3  in. 
apart,  will  be  required  for  a  fence  around  a  lot  16  rd. 
long  and  15  rd.  wide  ? 

37.  What  is  the  area  of  the  two  gables  of  a  barn  32  ft. 
wide,  the  ridge  being  9  ft.  higher  than  the  plates  ? 

38.  What  are  the  contents  of  a  triangular  piece  of  land, 
fhe  longest  side  of  which  is  141  rods,  and  the  perpendic- 
ular distance  to  the  angle  opposite  56  rods  ? 

39.  The  area  of  a  triangle  is  108  sq.  ft.,  and  its  altitude 
is  9  ft.     What  is  the  length  of  its  base  ? 

Jfi.  At  the  side  of  a  staircase  is  a  triangular  piece  of 
wall  18  ft.  long  and  13  ft.  high.  What  is  its  area,  in 
square  yards? 

Ji.1.  Find  the  area  of  a  cubical  block  whose  edge  meas- 
ures 10  inches. 

^2.  A  100 -acre  farm  is  a  trapezoid,  the  two  parallel 
sides  of  which  are  64.7  rd.  and  135.3  rd.  How  far  apart 
are  these  two  sides  ? 

43.  A  lot  of  land  lying  between  two  parallel  streets  3 
rods  apart,  measures  200  ft.  on  one  street  and  160  feet  on 
the  other.     How  many  sq.  rd.  does  it  contain  ? 

Jt^J^.  Find  the  cost,  at  $28  per  M,  of  l|-incli  matched 
flooring  for  a  two-story  house  32  by  24  ft.,  making  the 
customary  allowance  for  matched  lumber. 


234  SECOND    BOOK   IX  ARITHMETIC. 

^5.  Find  the  area  of  a  field  40  rd.  long,  15  rd.  wide  at 
one  end,  and  \  rd.  wide  at  the  other. 

Jfi.  What  is  the  volume  of  a  space  22.5  ft.  by  6.4  ft.  by 
3.25  ft.  ? 

^7.  Find  the  cubic  contents  of  a  marble  slab  6  ft.  long, 
2  ft.  G'  wide,  and  3^'  thick. 

JiS,  In  a  school  room  32  ft.  long,  24  ft.  wide,  and  12  ft. 
6'  high  are  60  pupils,  each  breathing  12 J  cu.  ft.  of  air  per 
hour.  In  wliat  time  will  they  breathe  as  much  air  as  the 
room  contains  ? 

Ji9,  In  digging  a  mill-race  120  rd.  long  and  30  ft.  wide, 
5,500  cu.  yd.  of  earth  were  removed.  What  was  the  depth 
of  the  race  ? 

50,  A  pile  of  4-foot  wood  24  rods  long  contains  100 
cords.     What  is  its  height  ? 

51,  How  many  more  bushels  of  barley  than  of  com  in 
the  ear,  can  be  stored  in  a  bin  12f  ft.  long,  8  ft.  wide,  and 
5.5  ft.  deep? 

52.  How  many  sheets  of  roofing  tin,  each  covering  14' 
by  22',  will  cover  3.5  squares  of  roof  ? 

53.  At  $  .25  per  sq.  yd.  for  plastering,  and  $  .50  per  roll 
for  paper-hanging,  what  will  it  cost  to  plaster  and  paper  a 
room  18  ft.  by  16  ft.  by  9.5  ft.,  the  paper  being  \  yd.  wide, 
and  9  yd.  in  a  roll ;  and  making  allowance,  in  papering,  for 
2  windows,  each  3  ft.  by  6  ft.,  and  3  doors,  each  3  ft.  6'  by 
7  ft.  6'? 

5J^.  A  sewer  2f  mi.  long,  5  ft.  wide,  and  12  ft.  deep,  cost 
$6,468.     What  did  it  cost  per  cubic  yard  ? 

55.  Find  the  cost  of  digging  a  cellar  27  ft.  by  19  ft.  by 
7  ft.,  at  $  .18  per  cubic  yard. 

56.  A  rock  excavation  20  ft.  long,  10  ft.  wide,  and  9  ft. 
deep  will  hold  how  many  barrels  of  water  ? 


REVIEW  PROBLEMS   IN  MEASUREMENTS.     235 

57.  What  depth  of  rain-fall  upon  a  roof  30  by  40  feet 
will  fill  an  80-hogshead  cistern  ? 

58.  How  many  bricks  are  in  the  walls  of  a  church  60  ft. 
long,  40  ft.  wide,  the  side  walls  28  ft.  high,  the  gable  peaks 
46  ft.  6^  high,  and  the  walls  1  ft.  thick ;  allowances  being 
made  for  one  half  of  the  openings,  which  are  {a)  one 
door-way  7  ft.  by  12  ft.,  (b)  two  door-ways  each  8  ft.  2^  by 
8  ft.  6^,  {c)  ten  windows  each  3  ft.  by  8  ft.  6^,  and  {d)  ten 
windows  each  3  ft.  by  7  ft.  6'? 

59.  Allowing  221-  bricks  to  make  a  cubic  foot  of  brick 
wall,  what  part  of  the  wall  consists  of  mortar  ? 

60.  At  33^^  per  sq.  yd.  for  labor,  and  $7.50  per  M  for 
brick,  what  will  a  contractor  receive  for  a  brick  pave- 
ment in  a  court  3  rd.  long  and  40  ft.  wide,  the  brick  being 
laid  flatwise  ? 

61.  At  $  .93f  per  perch  for  the  stone,  and  $1.25  per 
cubic  yard  for  the  mason  work,  what  is  the  cost  of  a  par- 
tition wall  between  the  cellars  of  two  stores,  the  wall  being 
72  ft.  long,  8  ft.  high,  and  18'  thick,  and  1  perch  of  stone 
making  22  cubic  feet  of  masonry  ? 

62.  Find  the  cost  of  9  pieces  of  floor  joists  13  ft.  long 
and  2'  by  9',  at  $14.50  per  M. 

63.  How  many  tons  are  there  in  a  stack  of  hay  22  ft. 
by  35  ft.  on  the  ground,  12  ft.  high  to  the  slope,  and  the 
ridge  of  the  roof -shaped  top — which  extends  lengthwise, — 
15  ft.  higher  than  the  sides  ? 

6"^.  How  much  1^'  flooring  will  be  required  for  a  school 
room  32  ft.  by  48  ft.,  and  a  class  room  -J  as  long  and  J  as 
wide  ? 

65.  At  $16.25  per  M,  what  must  I  pay  for  8  boards 
18  feet  long,  14  inches  wide  at  one  end,  and  tapering  to  a 
point  ? 


236  SECOND    BOOK  IN  ARITHMETIC. 

66.  I  -am  building  a  carriage  -  house  19  by  28  feet. 
How  mucli  will  the  2-inch  plank  for  the  floor  cost  me,  at 
$17.50  per  M  ? 

67.  A  wagon  box  is  9  ft.  9'  long,  3  ft.  4'  wide,  and  2  ft.  2' 
deep,  inside  measurements.  IIow  many  bushels  of  wheat 
will  it  hold,  if  filled  even  full  ?  IIow  many  bushels  of 
potatoes  ? 

68.  A  cellar,  21.5  ft.  by  29.5  ft.  has  a  cement  floor. 
How  much  did  it  cost,  at  $1.15  per  sq.  yd.  ? 

69.  "What  is  the  value  of  a  stick  of  ship  timber  32  ft. 
long  and  16'  by  20^  at  $18  per  C  ? 

70.  At  $1.1 8f  per  perch  for  stone  and  labor,  what  will 
be  the  cost  of  the  two  abutments  of  a  bridge,  each  24  ft. 
long,  11  ft.  wide  at  the  bottom,  7  ft.  wide  at  the  top,  and 
15  ft. high? 

71.  A  farmer  built  a  stone  fence  72  rd.  long,  3  ft.  6'  high, 
2  ft.' thick  at  the  bottom,  and  1  ft.  thick  at  the  top.  How 
many  cords  of  stone  did  he  use  ? 

72.  A  mow  of  hay  1  rd.  square  at  the  ends  and  33  ft. 
long,  contains  how  many  tons  ? 

73.  A  coal-house  15  ft.  by  8  ft.  by  7i  ft.  will  store  how 
many  bushels  of  coal  ? 

74.  At  40^  per  bunch  for  lath,  and  22^  per  square  yard 
for  lathing  and  plastering,  what  will  it  cost  to  lath  and 
plaster  this  room,  making  the  customary  allowance  for 
openings  ? 

75.  What  must  be  the  length  of  a  bin  6^  ft.  wide  and 
4|-  ft.  deep,  to  contain  2^  times  as  much  as  a  bin  7  ft.  long, 
5  ft.  wide,  and  3f  ft.  deep  ? 

76.  A  packing-box  made  of  1-inch  lumber  is  2  ft.  4J' 
wide  and  1  ft.  6^'  deep,  inside  measurement ;  and  its  ca- 
pacity is  12  cu.  ft.  587.1  cu.  in.  What  are  the  outside 
dimensions  of  the  box  ? 


CHAPTEK    VII. 

PERCENTAGE. 


SECTION    I. 


NOTATION. 


960*  The  term  per  cent  means  by  the  hundred. 

1  per  cent  is  one  of  every  hundred,  or  one  hundredth. 

6  per  cent  is  6  of  every  hundred,  or  6  hundredths. 

16|  per  cent  is  16|  of  every  hundred,  or  16|  hundredths. 

A.  How  many  hundredths  of  a  number 

1.  Is  1  per  cent  of  it  ?  «5.  5  per  cent  ? 

2.  Are  3  per  cent  of  it  ?      Jj..  ^  per  cent  ? 


^ .  8  per  cent  ? 
^.10  per  cent? 


-♦.   sw.'o  o  per  k:^u\)  ui.  II  "i        j^.    i   pt;i  ctui  ?         u.    ±kj  per  ce 

What  fractional  part  of  a  number,  in  its  lowest  terms, 
7.  Is  20  per  cent  of  it  ?     10.  GJ  per  cent  ?       IS.  100  per  c( 


/.  IS  20  per  cent  of  it? 

8.  Is  25  per  cent  of  it? 

9.  Is  50  per  cent  of  it  ? 

B. 

^.  iof 


10.  ej  per  cent . 

11.  121  per  cent? 

12.  331  per  cent " 


i.  "What  per  cent  of  any  number 
'it?       7.  4- of  it?        i^.  T^ofit? 

8.  I  of  it? 

9.  fof  it? 

10.  -5^  of  it? 

11.  tVof  it? 


—  ..  — ^ — , 

13.  100  per  cent? 
H.  140  per  cent? 
15.  1 1 6f  per  cent  ? 


o.  f  of  it? 

^.  iof  it? 

5.  I  of  it? 

^.  I  of  it? 

C  1.  What  decimal  part  of  a 
number  is  ^  per  cent  of  it  \ 
2.  J  per  cent  ?     5.  f  per  cent  ? 
^.  |-  per  cent 

*  •     in 


•J-  per  cent  ? 
-jiy-per  cent  ? 


is  1  of  it  ? 

17.  14  of  it? 

18.  A- of  it? 
IP.  it  of  it? 

20.  yVo  of  it? 

21.  The  whole  of  it? 

*.  What  fractional  part  of  a 
nnmlipr  is  i  per  Cent  of  it? 


12.  ^ig-of  it? 

13.  T^of  it? 

i^.  A  of  it? 

15.  -5^  of  it? 

16.  -/^of  it? 


number  is  ^ 
9.  iper  cent? 
i(?.  ^  per  cent  ? 


yV  per  cent  ?  ii.  -|-  per  cent  ? 


i^.  f  per  cent  ? 
i^.  |-  per  cent  ? 
i^.  f  per  cent? 


367.  The  conunercial  sign^  %^  signifies  per  cent. 
15^  signifies  15  hundredths,  and  is  read  "15  per  cent." 


238  SECOND    BOOK  IN  ARITHMETIC. 

Per  cent  may  be  applied  to  any  number. 
40  per  cent  of  1  bushel  =  .40  bu.  =  yV(>  t>u.  =  f  bu. 
14^"      •'     "365  days  =  .145,  or  ^Vif  of  365  days. 
7   *'      **     "  $85.42  =  .07,  or  -j^^  of  $85.42. 
16| "      "     ''  93}  =  .161,  or  ■i^%,  or  J  of  93}. 

125^  is  1.25,  or  f.       I     12J^  is  .12|-,  or  .125,  or -J. 

308^  is  3.08,  or  |-J.     |     {^  is  .25^,  or  .OOJ,  or  .0025.    Hence, 

968.  I.  To  expirees  per  cent  decimally^  two  decimal  fig- 
ures are  required. 

II.  To  express  100  per  cent  or  more^  an  integer  or  a 
mixed  decimal  number  is  required, 

III.  To  express  parts  of  1  per  cent^  either  decimal  fig- 
ures^ or  fractions  at  the  right  of  hundredths^  are  required. 


i^  or  .5^  is  .00^  or  .005. 
i^  or  .20^  is  .00^  or  .002. 


1^  or  .375^  is  .oof  or  .00375. 
ifo  or  .33^^  is  .00^. 


Write      j  8  per  cent  5  per  cent  40  per  cent      33  per  cent, 

decimally  (  3  per  cent  10  per  cent  14  per  cent      65  per  cent. 

Eeadandwrite  (  12^  10^    56,^  IG^fo      31.25^    200^ 

decimally       (    9^  90,^     64,^  18.7^    114^         J^     ^-^. 

Eead  as  per  cent    .07  .30     .19  1.37     2.038    .44|    .08^. 

S09.  JPercentage  is  tlie  process  of  finding  any  num- 
ber of  hundredths  of  a  number. 

In  this  process  three  elements  are  considered,  viz.,  the 
per  ce7itj  the  hase,  and  the  percentage. 

370.  The  i^^r  cent  is  the  decimal  that  expresses  the 
number  of  hundredths  of  the  base  to  be  found. 

27 1 .  The  base  is  the  number  the  hundredths  of  which 
are  to  be  found. 

S73.  The  2^^^(^^^^tage  is  the  result  obtained  by  find- 
ing the  required  number  of  hundredths  of  the  base. 

27tl.  The  amount  is  the  base  plus  the  percentage. 

$874,  The  difference  is  the  base  minus  the  percentage. 


SECTION  II. 


How  miicli  is 

3-  7.  1  per  cent "" 

'500? 

8-12.  4  per  cent 

150? 

13-17.  5  per  cent 

^of- 

70? 

18-22.  8  per  cent 

36? 

23-27.  6  per  cent . 

.325? 

i  S3.  5fo  of  140  cd.  of  wood. 
Find  •<  S4-.  Sfo  of  250  yd.  of  sheetings. 

(  55.  3^  of  50  thousand  shingles. 
•25, 
59-67.  The  base  is  ■{  60,   [  and  the  per  cent  is 


THE  FIVE  GENERAL   CASES  OF  PERCENTAGE. 
Case  I.  Base  and  per  cent  given,  to  find  percentage. 

Oral  Work.— 275.  1.  Howmucli  is  1%  of  50  bushels? 

7  per  cent  of  a  number  is  7  times  1  hundredth  of  it ;  1  hun- 
dredth of  50  bushels  is  .5  of  a  bushel,  and  7  times  .5  of  a  bushel 
are  3.5  bushels. 

2.  15^  of  80  boxes  of  raisins  are  how  many  boxes? 
15  per  cent  of  a  number  is  15  himdredths,  or  3  twentieths,  of 
it ;  and  3  twentieths  of  80  boxes  of  raisins  are  12  boxes. 

Plow  much  is 

28-32.  25fo    ^        r  400  acres? 
33-37.  12^fo  75  melons? 

38-42.  50^     lof-^  120  horses? 
43-4.7.   66f^  9  days? 

48-52.  SO fo    J        l  5.5  dollars? 

56.  12ifo  of  $75. 

57.  SOfo  of  190  mi. 

58.  Qi^  of  240  bu. 

^ '      /  What  is  the 

.7.5  J  (33i.)P«'''=''"*^g'=- 

^T0«  The  inrcentage  is  the  product  of  the  base  multiplied  hy 
the  per  cent. 

Problems. 

Written  Work. — 1.  8%  of  a  flock  of  475  sheep  are 
black.     How  many  black  sheep  are  in  the  flock  ? 

2.  At  a  school  of  125  pupils,  the  average  daily  attend- 
ance is  SSfo  of  the  whole  number.  What  is  the  daily  at- 
tendance ? 

3.  From  a  cask  of  44  gallons  of  oil,  7^fo  leaked  out. 
How  much  oil  leaked  out  ? 


240 


SECOND    BOOK  IN  ARITHMETIC. 


Jf..  62^^  of  a  townsliip  of  Western  land  is  prairie.  How 
many  acres  of  land  in  the  townsliip  are  prairie  ? 

5.  A  farmer  raised  9,875  bnshels  of  grain,  12Y/o  of  wliieli 
was  barley.     How  many  bushels  of  barley  did  he  raise  ? 

6.  llf^  of  an  army  of  63,000  men  are  riflemen.  How 
many  men  are  riflemen  ? 

7.  A  dealer  bought  6,250  tons  of  coal,  33^  of  which  was 
Lehigh,  46^  Lackawanna,  and  the  balance  Pittston.  How 
many  tons  of  each  kind  of  coal  did  he  buy  ? 

8.  A  man  thrashed  6,319  bushels  of  oats  for  12}^  of 
them.     How  many  bushels  of  oats  did  he  receive  ? 

Case  IL  Percentage  and  base  given,  to  find  per  cent. 

Oral  Work. — 077.  1.  3  is  what  per  cent  of  15  ? 

3  is  3  fifteenths  or  1  fifth  of  15.  Since  15  is  100  per  cent 
of  itself  1  fifth  of  15  is  1  fifth  of  100  per  cent,  or  20  per  cent. 


'2.  What  per  cent  of  18  is  9  ? 
S.  Of  25  is  10?  7.  Of  2  is  I? 
U,  Of  46  is  V?        8.  Of  i  is  I? 

5.  Of  120  is  12  ?    9.  Of  $72  are  $6  ? 

6.  Of  22.5  is  n\%  10,  Of  $52  are  $13  \ 
15.  Of  6  bn.  1  pk.  are  1  bu.  3  pk.  ? 

17-25.  The  base  is  Uj    [  ^^^  the  per- 

centage  is 


What  per  cent 

11.  Of  500  bu.  are  50  bn.  ? 

12.  Of  75  gal.  are  15 gal.? 
IS.  Of  30  mo.  are  10  rao.? 
U.  Of  380  ft.  are  19  ft.? 

16.  Of  f  h.  is  I  h.  ? 

^•^*  ]  What  is  the 
2,5*)     percent? 


37 8,  The  per  cent  is  the  quotient  of  the  percentage  divided 
by  the  base. 

Peoblems. 

Written  Work, — 1.  $15  per  acre  for  the  use  of  gar- 
den land  vahied  at  $250  per  acre,  is  what  %  on  the  value 
of  the  land  ? 

2.  25  bushels  of  oats  are  what  %  of  5,000  bushels  ? 


PER  CENT  A  GE.— GENERAL    CASES. 


241 


3,  A  farmer  harvested  270  bushels  of  potatoes  from  18 
bushels  of  seed.     The  seed  was  what  %  of  the  yield  ? 

4,  The  yield  was  what  %  of  the  seed  ? 

5,  A  sheep  grower  sold  95  sheep  from  a  flock  of  475. 
What  fo  of  the  flock  did  he  sell  ? 

6,  $119  are  what  %  of  $340  ? 

7,  A  house,  worth  $7,500,  rents  for  $600  a  year.    What 
%  on  its  value  does  it  rent  for  ? 

S.  A  fruit  grower  transplanted  250  peach-trees,  and  45 
of  them  died.     What  %  of  them  died  ? 

9.  975  acres  of  a  Southern  plantation  of  12,350  acres 
are  marsh.     What  %  of  the  plantation  is  marsh  ? 

10.  A  fruit  dealer  bought  a  cargo  of  85,744  oranges, 
and  lost  5,359  of  them.    What  %  of  the  cargo  did  he  lose  ? 

Case  III.  Percentage  and  per  cent  given,  to  find  base. 
Oral  Work. — S79, 1.  21  is  6  per  cent  of  what  number? 

21  is  6  per  cent  of  100  times  1  sixth  of  21 ;  1  sixth  of  21  is 
3.5;  and  100  times  3.5  are  350. 

^.  75  is  10  per  cent  of  what  number  ? 

Since  10  per  cent  of  any  number  is  1  tenth  of  the  nimiber,  75 
is  10  per  cent,  or  1  tenth,  of  10  times  75,  or  750. 

What  is  the  number  of  which 


s. 

5. 
6. 

10  is  40^? 
12  is  10^? 

9  is  33^-^? 

8  is  25,^? 

7. 

4is8|^? 

8:  2iisl6f^? 
P.     -J  is  50^  ? 

10.  I  is  75^? 

11.  $60  are  120^? 

12.  -J  mi.  is  175^? 
r  7.5,  J  and  the 

18-26.  The  percentage  is  •<  42,  >■  per  cent 
(  2i,  )        is 

280o  The  base  is  the  quotient  of  the  percentage  divided  by  the 
per  cent. 


13.  30  lb.  are  5^  ? 
U.  13  da.  are  26^? 

15.  25  T.  are  62^^? 

16.  I  rd.  is  .3^  ? 

17.  71  gal.  are  2j^? 
6.] 

8.  V  What  is  the  base? 


242  SJECOND    BOOK  IN  ARITHMETIC, 

Problems. 
Written  Work.—l,  A  merchant  vessel  has  169  tons 
of  hides  on  board,  which  are  13^  of  the  whole   cargo. 
How  many  tons  are  there  in  the  cargo  ? 

2.  The  distance  between  two  stations  on  a  railroad  is 
11.5  miles,  and  this  is  12|^^  of  the  whole  length  of  the 
road.     What  is  the  length  of  the  road  ? 

3.  A  gentleman  sold  his  house  and  lot  for  18^  above 
cost,  and  made  $500.     How  much  did  the  property  cost  ? 

4'.  Monday  my  sales  amounted  to  $237.50,  which  was 
12|-^  of  my  sales  for  the  week.  How  much  were  my 
sales  for  the  week? 

5,  A  grocer  sold  10  sacks  of  coffee,  which  was  ^fo  of 
his  whole  stock.     How  many  sacks  of  coffee  had  he  ? 

6,  This  year  a  planter  sold  his  cotton  at  9  cents  a 
pound,  which  was  80^  of  the  price  he  received  last  year. 
At  what  price  did  he  sell  his  crop  last  year  ? 

7,  A  certain  school  closed  its  winter  term  with  115 
pupils,  which  was  92^  of  the  number  with  which  the  term 
began.     With  how  many  pupils  did  it  begin  ? 

8,  7.5  sq.  rd.  of  land  are  6%  of  how  many  square  rods  ? 

9,  $  .30^  are  120^  of  how  much  money  ? 

10.  i  lb.  of  tea  is  7^%  of  a  package  of  how  many  pounds  ? 

Case  IV.  Base  and  per  cent  g^iven,  to  find  amount 
or  difference. 

Oral  Work. — 381.  1.  The  base  is  25,  and  the  per  cent 

is  20.    What  is  the  amount  ?       2.  What  is  the  difference  ? 

For  the  amount — 25  plus  20  per  cent  of  25  is  120  per  cent 
of  25,  or  6  fifths  of  25,  which  is  30. 

For  the  difference.— 25  minus  20  per  cent  of  25  is  80  per  cent 
of  25,  or  4  fifths  of  25,  which  is  20. 


PERCENTAGE.— GENERAL    CASES.  243 

What  is  tlie  amount,  and  what  is  the  difference, 
J.  When  the  base  is  60,  and  the  per  cent  is  15  ? 
^.  The  base  being  14  mo.,  and  the  per  cent  25  ? 

5.  The  per  cent  being  33^,  and  the  base  27  cu.  ft.  ? 

6.  If  the  base  is  $6.4,  and  the  per  cent  is  6;|-  ? 

7.  When  -J  yd.  is  the  base,  and  8-|^  the  per  cent  ? 

8-2S.  Given,  J  hk'     (    and  the    )    oa!  (.  to  find  j  amount, 
the  base      /  i  05   \    P^^  ^^^^    /  1 40 '  \      ^^^^     f  <iifference. 

^83,  I.  TAe  amount  is  the  product  of  the  base  multiplied  by 
1  plus  the  per  cent ;  and 

II.  The  difference  is  the  product  of  the  base  multiplied  by  1 
minus  the  per  cent. 

Problems. 

Written  Work, — 1,  If  oak  bark  is  worth  25^  more 
than  hemlock,  and  hemlock  sells  at  $6.72  a  cord,  how 
much  is  a  cord  of  oak  bark  worth  ? 

^.  A  lady  having  $600,  paid  18^  of  it  for  a  parlor 
organ.     How  much  money  had  she  left  ? 

3.  Last  year  a  merchant  bought  table  linens  at  60^  a 
yard,  but  this  year  he  pays  37|-^  more  for  the  same  class 
of  goods.     How  much  does  he  pay  per  yard  this  year  ? 

Jf.,  A  gentleman  having  $876  deposited  in  a  bank,  check- 
ed out  62J^  of  it.     How  much  remained  on  deposit  ? 

5.  If  a  man's  income  is  $800,  and  he  expends  66f^  of 
it,  how  much  does  he  save  ? 

Case  Y.  Amount  or  diflference  and  per  cent  given, 
to  find  base. 

Oral  Work. — S83.  1.  The  amount  is  30,  and  the  per 

cent  is  20.     What  is  the  base  ? 

Since  30 — the  amount — is  100  per  cent  plus  20  per  cent,  or 
120  per  cent  of  the  base,  the  base  is  the  quotient  of  30  divided 
by  1.20  or  -f,  which  ic  2.5. 


244  SECOND    BOOK  IN  ARITHMETIC. 

£.  The  difference  is  24,  and  the  per  cent  is  25.  What 
is  the  base  ? 

Since  24 — the  difference — is  100  per  cent  minus  25  per  cent, 
or  75  per  cent  of  the  base,  the  base  is  the  quotient  of  24  di- 
vided by  .75  or  -|,  which  is  32. 

^  3.  Amount,  36;  per  cent,  12^;  ^ 

^1.  14'  Amount,  69;  per  cent,  15;      I  ,    ^    ,  ^,     , 

Given,  i  1:     .  /  .HI  .   o^      r  to  find  the  base. 

o.  Amount,  17^;  per  cent,  2o; 

.  6.  Amount,  $6.8 ;  per  cent,  6^- ;  J 

7,  What  number  of  cows,  plus  16%  of  the  number, 
equals  87  cows  ? 

8,  How  much  money,  less  78^  of  itself,  leaves  $44  ? 

9,  How  many  yards,  less  5^  of  the  number,  equals  57 
yards  ? 

10.  What  number,  plus  7%  of  the  number,  equals  321  ? 

11.  Any  amount  is  the  base,  or  100  per  cent,  plus  what  ? 

12.  Any  difference  is  the  base  minus  what  ? 

S84,  The  base  is  the  quotient 

1.  Of  the  amount  divided  by  1  plus  the  per  cent;  or 

2.  Of  the  difference  divided  by  1  minus  the  per  cent. 

Problems. 
Written  Work, — 1.  By  selling  a  piano  for  $540, 1 
gain  20  per  cent.     What  did  it  cost  me  ? 

2.  Sold  a  carriage  for  $270,  which  was  20^  less  than  it 
cost.     What  was  the  cost  ? 

3.  This  year  a  lawyer's  receipts  from  his  practice  are 
$2,662.50,  which  is  42^  more  than  they  were  last  year. 
How  much  were  his  receipts  last  year  ? 

Jf..  A  physician  saves  $56  a  month,  and  his  expenses 
are  60^  of  his  yearly  income.     What  is  his  income  ? 

5.  My  income  is  %%  greater  this  year  than  it  was  last, 
and  this  year  it  is  $1,890.     How  much  was  it  last  year? 


TERCENTAGE.— GENERAL    CASES.  245 

6,  A  manufacturer  sells  reapers  at  $126  apiece,  and 
gains  40^.     How  mucli  do  tliey  cost  him  apiece  ? 

7.  44  yards  of  cloth  measured  2^%  more  before  being 
sponged  than  after.  What  was  its  length  after  being 
sponged  ? 

^85,  KrLES  FOR  Peecentage. 

I.  Base  and  per  cent  given,  to  find  percentage:— 
Multijply  the  hase  hj  the  jper  cent, 

II.  Percentage  and  base  given,  to  find  per  cent:— 
Divide  the  percentage  hy  the  hase. 

III.  Percentage  and  per  cent  given,  to  find  base:— 
Divide  the  jperGentage  hy  the  per  cent. 

lY.  Base   and  per   cent   given,  to  find  amount  or 
difference  :— 

1.  For  the  amount : — Multiply  the  hase  hy  1  plus  the 
per  cent. 

2.  For  the  difference  :--i/t^?^^J!?Zy  the  hase  hy  1  mimes 
the  per  cent. 

V.  Amount   or   difference    and  per   cent  given,  to 
find  base:— 

1.  Divide  the  amount  hy  1  plus  the  per  cent;  or 

2.  Divide  the  difference  hy  1  minus  the  per  cent. 

a.  The  terms  base,  per  cent,  and  percentage  correspond  to 
the  terms  multipUcand,  multiplier,  and  product.     Hence, 

Any  two  of  the  three  terms — base,  per  cent,  percentage — 
being  given,  the  third  term  may  be  found.     Thus : 

he  In  multiplication  c.  In  percentage 

I.  Multiplicand  X  multiplier  =  product.  Hence,  I.  Base  X per  cent  =  percentage. 

Product  1  .  1.       1  TT  l^€'^'(^c'>'^l<^9^ 

II. ; — - —  =  multiplicand.  Hence,  IL =  base. 

multiplier  per  cent 

Product  ,     ,  ^,^  Parentage 

III. —-7: 7  =  multiplier.  Hence,  III. ; —per  cent. 

multiplicand  base 


SECTION  III. 

SPECIAL  APPLICATIONS  OF  THE  FIVE  GENERAL  CASES. 
I.  PROFIT   AND   LOSS. 

^86,  Profit  is  the  sum  above  cost  for  which  goods 
are  sold. 

387.  Loss  is  the  sum  below  cost  for  which  goods  are 
sold. 

In  computations  in  profit  and  loss, 

a.  Costz=z  base;       \       &.  Profit  or  loss  =  percentage ; 
c.  Selling  price  =  amount  or  difference. 

Problems. 

1,  10%  profit  on  $2.50  cost,  is  how  much  profit  ? 

^.  8^  loss  on  $7.25  cost,  is  how  much  loss  ? 

3.  I  bought  vinegar  at  $  .16  a  gallon,  and  sold  it  at  a 
profit  of  56^^.     How  much  did  I  gain  on  a  gallon? 

^.  A  man  sold  a  city  lot  that  cost  him  $160,  at  an 
advance  of  212^^.     How  much  did  he  gain  ? 

6.  I  bought  a  carriage  for  $187.50,  and  sold  it  at  a  loss 
of  12|-^.     How  much  did  I  lose  ? 

6.  If  a  milliner  sells  ribbon  that  cost  $  .31}  a  yard,  at 
a  profit  of  20^,  how  much  does  she  gain  on  a  yard  ? 

7.  Cost,  $  .40 ;  profit,  62^^.    What  is  the  selling  price  ? 

8.  Cost,  $1.20 ;  loss,  33^^.     What  is  the  selling  price  ? 

9.  The  bread  made  from  a  barrel  of  flour  weighs  A.0% 
more  than  the  flour.     How  many  pounds  does  it  weigh  ? 

10.  A  grocer  sold  tea  that  cost  him  $  .78  a  pound,  at  a 
loss  of  15^.     At  what  price  per  pound  did  he  sell  it  ? 

11.  Mark  silk  that  cost  $2.60  per  yard,  to  sell  at  20^  loss. 


FEBCENTAGE.^GENERAL   CASES.  2-17 

1£.  Mark  silk  that  cost  $2.60  per  yard,  to  sell  at  25%  gain. 
13.  Mark  gold  pens  that  cost  $1.25,  to  sell  at  60%  profit. 
14-.  Mark  kid  gloves  that  cost  $  .93f ,  to  sell  at  9ifo  loss. 

15.  $  .87i  profit  on  $10.50  cost,  is  what  per  cent? 

16.  $  .02|-  loss  on  $  .15  cost,  is  what  per  cent  ? 

17.  A  flour  dealer  bought  flour  for  $6.56  a  barrel,  and 
sold  it  for  $7.31.     What  %  did  he  make  ? 

18.  What  ^  do  I  make,  by  selling  eggs  at  f  of  their  cost  ? 

19.  If  kerosene  is  bought  at  f  of  the  market  price,  and 
sold  at  10^  below  the  market  price,  what  %  is  lost  ? 

W.  I  sell  for  $5  what  cost  me  $4.25.    What  ^  do  I  gain  ? 

21.  I  sell  for  $  .30  what  cost  me  $  .37^.   What  ^  do  I  lose? 

22.  A  jeweler  sold  a  watch  for  $112.50,  w^hich  was  15^ 
advance  on  the  cost.     How  much  was  the  cost  ? 

23.  A  gentleman  sold  a  horse  and  harness  for  $187.50, 
which  was  10^  less  than  cost.     How  much  was  the  cost  ? 

2^.  $  .90,  the  retail  price  of  a  book,  is  4:0%  above  the 
wholesale  price.     What  is  the  wholesale  price  ? 

II.   COMMISSION. 

388.  An  agent  is  a  person  authorized  to  transact 
business  for  another. 

S89.  A  commission-merchant  is  a  merchant  who 
buys  or  sells  goods  or  other  property  as  an  agent  for  others. 

390.  Cofumission  is  a  percentage  paid  to  an  agent 
for  transacting  business. 
In  computations  in  commission, 

a.  Sum  expended  or  collected  hy  agent  ^=z  base; 

h.  Commission  ==  percentage  ; 

c.  Sum  collected  or  expended^  plus  commission  =  amoimt. 


248  SECOND    BOOK   IX  ARITHMETIC. 

Problems. 

1.  5%  on  $415  collections,  is  how  much  commission  ? 

^.  2^%  on  $8,362-j%-  invested,  is  how  much  commis- 
sion ? 

3.  A  real-estate  agent  sold  a  house  and  lot  for  $2,275. 
How  much  was  his  commission,  at  2%  ? 

^.  How  much  commission  will  an  auctioneer  receive 
for  selling  a  stock  of  goods  for  $1,975,  at  Sfo  ? 

5.  An  agent  buys  8,040  lb.  of  wool,  at  $  .37|-  a  pound. 
How  much  is  his  commission,  at  2^%  ? 

6.  I  sold  15,000  yards  of  sheetings,  at  $.11^.  How 
much  was  my  commission,  at  1^%  ? 

7.  $29.80  for  collecting  $238.40,  is  at  what  per  cent? 

8.  $56.33  for  investing  $2,319,  is  at  what  per  cent  ? 

9.  $1,390  for  investment  after  deducting  5%  commis- 
sion, is  how  much  for  investment  ?  How  much  for  com- 
mission ? 

10.  $12,400  includes  investment,  and  commission  at  S^%, 
What  is  the  investment  ?     What  is  the  commission  ? 

11,  A  druggist  sent  his  broker  or  agent  $2,630  with 
which  to  buy  goods,  after  deducting  his  commission  of 
2^fo'     How  much  did  the  broker  expend  for  goods  ? 

12,  A  wool  buyer  receives  $5,600  with  which  to  pur- 
chase wool,  less  2%  commission  on  the  money  paid  out. 
How  much  money  will  he  expend  for  wool  ? 

13.  A  broker  received  $6,500  with  which  to  buy  hops, 
at  $  .31;|-  a  pound,  after  deducting  his  commission  of  i%. 
How  many  pounds  of  hops  did  he  buy  ? 

On  what  amount  (  1^.  Is  $41. 70  the  commission,  at  15^? 

of  sales  (  15.  Is  $450.30  the  commission,  at  7|-^? 

16.  A  collector  received  $123.75  for  collecting  bills,  at 
6%  commission.     How  much  money  did  he  collect  ? 


PERCENT AGE.-^QENERAL    CASES.  249 

III.  INSURANCE. 
^91«  Insurance  is  a  secnrity  against  loss  or  damage. 
Insurance  is  of  various  kinds, — as  Jire  insurance,  marine  in- 
surance, life  iyisurance,  health  insurance,  accident  insurance. 

^9^.  Valuation  is  the  sum  contracted  to  be  paid  to 
the  party  insm'ed,  for  property  destroyed  or  damaged. 

393.  JPremiiim  is  the  sum  paid  for  the  insurance. 

^94.  A  policy  is  a  written  contract  between  the  in- 
surer and  the  insured. 

In  computations  in  insurance, 

a.  Valuation  =^  base ;      \       h.  Premium  z=. percentage. 

Problems. 

1.  4:%  on  $2,250  valuation,  is  how  much  premium? 

^.  '\%  on  $6,287  valuation,  is  how  much  premium? 

3.  What  premium  must  a  grocer  pay  for  a  policy  of 
insurance  of  $1,950  on  his  stock  of  goods,  at  IJ^? 

Jf.  At  \%  a  year,  what  premium  do  I  pay  yearly  for  an 
insurance  of  $3,750  on  my  house  ? 

5.  What  is  the  annual  premium  for  insuring  a  steam 
saw-mill  for  $2,250,  at  3|^? 

6.  Find  the  premium  on  a  cargo  of  flour,  shipped  from 
New  York  to  Liverpool,  insured  for  $9,000,  at  ^%. 

7.  An  ocean  steamship  is  insured  for  $97,500,  at  ^%  a 
voyage.     What  is  the  premium  per  voyage  ? 

At  what  (  8.  Is  $31.25  the  premium  on  $12,500  valuation  ? 
per  cent  (  9.  Is  $97.50  the  premium  on  $6,500  valuation? 

10.  A  shipper  paid  $37.50  for  an  insurance  of  $7,500 
on  a  cargo  of  produce.     What  %  premium  did  he  pay  ? 

11.  An  agent  took  a  risk  of  $2,500  on  a  stock  of  stoves, 
and  received  $50  premium.     What  was  the  per  cent  ? 

L2 


250  SECOND    BOOK  IN   ARITHMETIC. 

1^,  At  f  ^,  $9.50  is  the  premium  on  what  valuation  ? 
13,  A  premium  of  $28.12J,  at  |^,  is  on  what  valuation  ? 
14"  An  agent  received  $9.75  for  insuring  a  barn  and  its 
contents,  at  ^%.    What  was  the  valuation  of  the  property  ? 

IV.  TAXES. 

995.  Taxes  are  sums  of  money  levied  upon  persons 
and  property,  to  meet  public  expenses. 

A  property  tax  is  a  tax  on  personal  property  and  real- 
estate  ;  a  poll  tax  is  a  tax  on  the  person. 
In  computations  in  taxes, 

a.  Valuation  =  base  ;     |     b.  Tax  =z  percentage. 

Problems. 
1,  Find  the  tax  on  property  valued  at  $2,875,  at  1.3^. 
^.  What  is  the  tax  on  $932  of  valuation,  at  Afd 

3.  $3.57  tax  on  $277-^^%  valuation,  is  at  what  per  cent  ? 

4.  The  tax  on  $3,200  is  $12.    What  is  the  per  cent  ? 

5.  The  taxable  property  in  an  incorporated  village  is 
valued  at  $500,000,  and  a  tax  of  $1,875  is  voted  for  school 
purposes.     What  is  the  per  cent  of  the  tax  ? 

6.  In  the  same  village  A's  property  is  assessed  at  $1,250, 
B's  at  $1,500,  C's  at  $2,250,  D's  at  $750,  and  E's  at  $4,250. 
Allowing  5^  for  collector's  fees,  how  much  tax  has  each 
one  to  pay  ? 

7.  My  tax  is  $15.25,  at  lyf^-^-     Wliat  is  the  valuation  ? 

8.  What  is  the  assessed  value  of  a  farm  that  is  taxed 
$23.37|,  at  2|-  mills  on  the  dollar? 

9.  The  capital  of  a  glass  factory  is  owned  by  8  part- 
ners, whose  shares  are  $3,300,  $450,  $1,200,  $2,250,  $750, 
$1,800,  $600,  and  $900 ;  and  repairs  are  made  that  cost 
$225.  At  what  per  cent  must  the  capital  be  taxed,  to 
pay  for  the  repairs  ?     How  much  must  each  partner  pay  ? 


FERVENT AQE.—QENERAL    CASES.  251 

V.   CUSTOMS  OR  DUTIES. 
^96.  Customs  or  duties  are  taxes  levied  on  im- 
ported goods  and  otlier  property,  for  the  support  of  the 
General  Government. 

a,  Imports  are  goods  and  other  property  brought  into  a 
country. 

b,  A  custom-house  is  an  office  at  which  duties  are  collected. 

c,  A  port  of  entry  is  a  seaport  in  which  a  custom-house  is 
situated. 

SOT,  A  tariff  is  a  list  or  schedule  of  the  legal  rates 
of  duties  on  imports. 

398,  Duties  are  of  two  kinds,  ad  valorem  and  specific. 
a.  Ad  valor evfi  duties  are  duties  on  the  net  cost  of  im- 
ports, at  the  place  where  they  are  purchased. 
&.  Specific  duties  are  duties  on  the  number  or  quantity. 

^99.  Tare^  leakage^  and  breakage  are  deductions 
made,  on  certain  kinds  of  goods,  before  specific  duties  are 
computed. 

a.  Tare  is  a  deduction  for  the  weight  of  the  box,  cask,  bag, 

or  case  that  contains  the  goods. 
h.  Leakage  is  a  deduction  for  waste  of  liquors  in  casks  or 

barrels. 
c.  Breakage  is  a  deduction  for  loss  of  liquors  in  bottles. 

300.  Gross  weight  is  the  weight  without  deductions. 

301.  JS'et  weight  is  the  weight  less  the  deductions. 

30^.  An  invoice  is  a  WTitten  list  of  merchandise, 
with  prices  and  charges  annexed. 

A  manifest  is  a  complete  invoice  of  a  ship's  cargo. 
303.  In  computations  in  duties, 

a.  Net  value  =  base;     \     h.  Duty  —  'percentage. 


252  SECOND    BOOK   IX   ARITHMETIC, 

Rules  for  Computing  Duties. 

I.  For  ad  valorem  duties : — Mnltijply  the  net  value  of 
the  imports  hj  the  tariff  per  cent  of  duty. 

II.  For  specific  duties: — Multiply  the  tariff  rate  of 
duty  on  a  unit  hy  the  net  number  of  units. 

Problems. 

1.  The  gross  weight  of  225  boxes  of  raisins  is  33^  lb. 
per  box,  and  the  tare  is  25^.  What  are  the  duties,  at  2J^ 
per  lb.  ? 

^.  What  are  the  duties  on  316  boxes  of  lemons,  invoiced 
at  $3.45  per  box,  at  20^  ad  valorem  ? 

3.  What  are  the  duties  on  25  pieces  of  Binissels  carpet- 
ing of  65  yd.  each,  invoiced  at  $.43|  per  yd.,  the  tariff 
rates  being  $.44  per  yd.  specific  and  35^  ad  valorem  ? 

^.  A  sugar  refiner  imports  72  hlid.  of  W.  I.  sugar,  gross 
weight  975  lb.  each,  tare  12J^  invoice  price  6^^  per  lb. 
The  tariff  rates  are  2^  per  lb.  specific  and  25^  ad  valorem. 
What  are  the  custom-house  charges  ? 

VI.  STOCKS. 

304.  Stock  is  the  money  or  other  property  invested 
in  the  business  of  a  corporation. 

305.  A  share  of  stock  is  one  of  the  equal  parts  into 
which  the  stock  of  a  corporation  is  divided. 

The  original  value  of  a  share  is  commonly  $100. 

306.  The  par  value  of  stock  is  100  per  cent. 

307.  The  marUet  value  of  stock  is  the  sum  for 

which  it  will  sell. 

a.  Stock  is  at  par,  when  its  market  value  is  100  per  cent; 
h.  Stock  is  above  par,  when  its  market  value  is  above  100 

per  cent;   and 
c.  Stock  is  heloiv  par,  when  its  market  value  is  below  100 

per  cent. 


PERCENTAGE.— GENERAL    CASES.  25^ 

308.  Premium  is  the  excess  above  100  per  cent  in 
the  value  of  stock  that  is  above  par. 

309.  Discount  is  the  deficiency  below  100  per  cent 
in  the  value  of  stock  that  is  below  par. 

310.  A  stock-broker  is  an  agent  who  buys  and  sells 
stocks  for  others. 

311.  IBrokerage  is  the  commission  paid  to  stock- 
brokers. 

In  computations  in  stocks, 
a.  Par  value  =:  base;  |  b.  Premium  or  discount  :=z percentage ; 
c.  Market  value  —  amount  or  difference. 

Pkoblems. 
What  is  the  market  j  1,  That  sells  at  7f  ^^  premium  ? 
value  of  stock       |  2.  That  sells  at  16^  discount  ? 

V   rl  f Ti    i  ^'  l^^d^hmi  on  7  shares  of  stock,  at  5-J^  above  par. 
(  Jf.  Discount  on  35  shares  of  stock,  at  ^-}-^fo  below  par. 

5.  If  Western  Union  Telegraph  stock  is  at  a  premium 
of  15^,  how  much  must  be  paid  for  75  shares  ? 

6.  $8,982^^  buys  how  much  stock,  at  103^ ;  i.  e.^  at 
Z\%  premium  ? 

7.  $2,346,871-  buys  how  much  stock,  at  93|-;  i.  e.^  at 
<d^%  discount  ? 

8.  I  sell  telegraph  stock  at  par  that  cost  me  94.    What 
^  do  I  gain  ? 

9.  I  sell  stock  at  75  that  cost  me  80.    What  ^  do  I  lose  ? 

10.  What  is  the  market  value  of  50  shares  of  mining 
stock,  that  sells  at  32^^  below  par  ? 

11.  I  bought  15  shares  of  Novelty  Iron  Works  stock,  at 
116|-.     How  much  did  it  cost  me  ? 

12.  A  widow  invested  $787|-  in  toll -bridge  stock,  at 
37J^  below  par.     How  many  shares  did  she  buy  ? 


254  SECOND    BOOK   IN  ARITHMETIC. 

VII.  PARTNERSHIP. 

319.  A  partnershij^  or  company  is  an  associa- 
tion of  persons  for  the  transaction  of  business. 

A  fir^n  is  the  name  under  which  a  company  transacts  busi- 
ness.    A  firm  is  also  called  a  house, 

313.  Partners  are  the  persons  associated  in  a  part- 
nership or  company. 

a.  An  active  pawner  is  one  who  takes  an  active  part  in 
the  management  of  the  business. 

b,  A  silent  partner  is  one  who  furnishes  capital,  but  takes 
no  active  part  in  the  management  of  the  business. 

c.  A  general  partner  is  one  who  is  responsible  for  the 
debts  of  the  company,  to  the  amount  of  his  entire  property. 

d,  A  special  partner  is  one  whose  responsibility  is  limited 
to  a  certain  amount,  specified  in  the  written  articles  of 
partnership. 

314.  Capital  or  stoeh  is  the  money,  or  its  equiva- 
lent, invested  in  business. 

a.  Capital  may  be  money,  real  estate,  personal  property,  time, 

labor,  or  skill. 
6.  The  resources  or  assets  of  a  company  are  its  entire 

property,  including  capital,  and  all  demands  in  its  favor. 

c.  The  liabilities  or  obligations  of  a  company  are  its  en- 
tire indebtedness,  or  all  demands  against  it. 

d.  Net  capital  or  surplus  is  the  excess  of  resources  over 
liabilities. 

e.  A  deficit  is  the  excess  of  liabilities  over  resources. 

/.  When  resources  exceed  liabilities,  the  company  is  solvent; 
and  when  liabilities  exceed  resources,  the  company  is  in- 
solvent  or  bankrupt, 

SI 5.  JDividends  are  the  profits  divided  among  the 
partners  of  a  company. 


PERCENTAGE.--GENERAL    CASES.  255 

316,  Assessments  are  sums  to  be  paid  by  the  part- 
ners of  a  company,  to  meet  expenses  or  cover  losses. 

317,  Each  partner's  share  of  a  dividend  or  assessment 
is  such  a  ^  of  his  share  of  the  capital,  as  the  entire  divi- 
dend or  assessment  is  %  of  the  entire  capital. 

Ex.  M,  JN",  and  E  are  partners ;  M  furnishes  $3,500  of 
the  capital,  N  $2,500,  and  E  $2,000 ;  and  their  profits  are 
$3,200.     What  is  each  partner's  share  1 
Full  Solution. 
$3^200,  entire  dividend. 
$3,500 -\-  $2,500 -\-  $2,000  =  $8,000,  entire  capital 
$3,200-^  $8,0  00  =  .J,.0=:  40  per  cent  of  dividend, 
JfOfc  of  $3,500  =  $1,^0  0,  M's  share. 
JfOfo  of  $2,500  z=^$l,000,N's     " 
Jt-Ofo  of  $2,000  =  $      800,  R's      " 

318,  EuLE  FOR  Partnership. 

I.  For  per  cent  of  dividend  or  assessment : — Divide  the 
total  dividend  or  assessment  hy  the  total  capital. 

II.  For  each  partner's  dividend  or  assessment : — Multi- 
ply his  capital  hy  the  per  cent  of  dividend  or  assessment. 

Problems. 

1.  Eeed  and  Clark  form  a  partnership.  Eeed  furnishes 
$3,500  of  the  capital,  and  Clark  $5,000.  They  gain  $2,450. 
What  is  each  partner's  share  ? 

2.  A,  B,  and  C  hire  a  pasture  for  $22.50.  A's  horse  is 
in  the  pasture  5  months,  B's  horse  7  months,  and  C's  horse 
6  months.     How  much  of  the  rent  does  each  pay  ? 

S.  A  carpet  factory  was  damaged  by  fire  to  the  amount 
of  $15,180,  and  it  was  insured  in  the  Hanover  Ins.  Co.  for 
$9,000,  in  the  Globe  Ins.  Co.  for  $7,200,  in  the  ^tna  Ins. 
Co.  for  $6,000,  and  in  the  Franklin  Ins.  Co.  for  $5,400. 
How  much  of  the  loss  did  each  company  sustain  ? 


256  SECOND    BOOK  IN  ARITHMETIC, 

Jf,.  Bates  and  Davis  lose  $828  in  trade.  Bates's  capital 
is  $1,200,  and  Davis's  $1,600.     What  is  the  loss  of  each? 

6.  The  total  capital  of  a  firm  is  $75,000,  of  which 
G's  share  is  |  less  than  H's,  and  K's  is  one  half  the  sum 
of  G's  and  H's.  The  year's  profits  are  $18,600.  Ee- 
quired,  each  partner's  capital,  and  each  partner's  dividend. 

6.  Four  men  shipped  6,000  bales  of  cotton  to  England, 
E  furnishing  2,100  bales,  F  1,250  bales,  G  1,725  bales, 
and  II  the  balance.  In  a  storm  1,920  bales  were  thrown 
overboard.     What  was  each  man's  share  of  the  loss  ? 

7,  Williams,  Jones,  and  Brown  manufacture  parlor 
organs.  Mr.  W.  furnishes  $3,000  of  the  capital,  Mr.  J. 
$6,750,  and  Mr.  B.  $8,250,  and  in  four  years  their  profits 
are  $70,200.     What  is  each  partner's  share  ? 

S.  Two  men  harvest  and  thrash  945  bushels  of  wheat 
for  -J-  of  the  crop,  A  furnishing  8  men  7  days,  and  B  14 
men  5  days.  How  many  bushels  of  wheat  does  each  part- 
ner receive  ? 

9,  A,  B,  and  C  own  100  shares  of  ER.  stock ;  and  A's 
annual  dividend  is  $210,  B's  $400,  and  G's  $240.  How 
many  shares  of  stock  does  each  own  ? 

10.  Hilton  and  Eoberts  hire  a  pasture  for  $50.  Hilton 
puts  in  25  horses  30  days,  and  Eoberts  puts  in  20  horses 
42  days.     How  much  of  the  rent  ought  each  to  pay  ? 

11,  January  1,  A,  B,  and  C  begin  the  manufacture  of 
hats,  with  a  capital  of  $20,400,  -^  of  which  is  A's,  f  is 
B's,  and  the  remainder  is  G's.  May  1,  C  buys  \  of  B's 
share,  and  D  buys  |  of  A's  share.  The  profits  for  the 
year  are  $8,420.     How  much  is  each  partner's  share  ? 

1^.  A,  B,  C,  and  D  formed  a  partnership.  A  furnished 
4  times  as  much  of  the  capital  as  B,  B  |-  as  much  as  C, 
and  D  as  much  as  A  and  B  together.  Their  profits  were 
$1,943.76.     What  was  each  partner's  dividend  ? 


[257  J 


SECTION  IV. 

INTEREST. 

310.  Interest  is  the  sum  paid  for  the  use  of  money. 

3^0.  JPrineipal  is  the  money  for  the  use  of  which 
interest  is  paid. 

3^1.  Amount  is  the  sum  of  principal  and  interest. 

3^^.  The  rate  of  interest  is  the  number  of  hun- 
dredths allowed  for  the  use  of  $1  of  principal  for  1  year. 

The  2^er  cent,  in  interest,  depends  upon  the  rate  of  interest 
and  the  time. 

3^3.  In  computations  in  interest, 

a.  Principal  =  the  base ; 

b.  Rate  of  interest  x  time  (in  years)  z=zper  cent; 

c.  Interest  =  the  percentage.     Hence, 

Interest  =  principal  x  rate  of  interest  x  time;  i.e., 
Interest  —  jprincijpal  y.  jper  cent. 

Case  I.  Interest  for  years. 

Oral  Work. — 3^4.  A.  At  6  per  cent  per  annum,  the 
rate  of  interest  on  $1  for  1  year  is  .06  or  j^. 

What  is  the  per  cent  on 


7.  $10  for  7  yr.? 

8.  $100  for  1  yr.? 

9.  $100  for  2  yr.? 


1.  $1  for  3  yr.  ?         4-  $1  for  8  yr.  ? 

2.  $3  for  1  yr.  ?         6.  $1  for  4  yr.  ? 

3.  $3  for  5  yr.  ?         6.  $5  for  4  yr.  ? 

J5.  At  6  per  cent  per  annum,  the  interest  of  $1  for  1 
year  is  $  .06. 
What  is  the  interest  of 


1.  $1  for  7  yr.  ? 

2.  $8  for  7  yr.  ? 

3.  $1  for  6  yr.  ? 


4.  $50  for  6  yr.  ? 

5.  $15  for  10  yr.? 

6.  $300  for  4  yr.  ? 


7.  $.50  for  1  yr.? 

8.  $  .50  for  5  yr.  ? 

9.  $1.50  for  5  yr.? 


258 


SECOND    BOOK  IN  ARITHMETIV. 


Formulas. 


(    I.  Int.  for  1  yr.  z=:  prin.  x  rate  of  int. 

\  II.  Int.  for  i/r.=zprin.  X  rate  of  mt.  for  No.  of  yr. 


Written  TForA?.— Ex.  What  is  the  interest  of  $135.25 
for  3  years,  at  ^%  ? 

Explanation. — \»t.  For  1  year.  1 
multiply  $135.25,  the  principal,  by 
.06,  the  rate  of  interest,  and  obtain 
$8.  Hi 

2d  For  Z  years.  I  multiply  $8.11  i^, 
the  interest  for  1  year,  by  3,  and 
obtain  $24.34^,  the  required  in- 
terest. 


Process. 
$135,25    Principal. 
.06    Rateofint. 

$8.1150    Int.forlyr. 
^ 

$2  4"  3  4- 5        Int.  for  3  yr. 

Problems. 

1.  What  is  the  interest  of  $467  for  1  year,  at  6^? 

2.  Find  the  interest  of  $321  for  1  year,  at  5^.     Find 
the  amount. 

3.  Find  the  interest  of  $167.50  for  6  years,  at  7%. 

U.  What  is  the  interest  of  $612.75  for  3  years,  at  %% 

5.  What  sum  of  money  will  pay  a  debt  of  $165.88, 
1  year  after  it  is  due,  with  interest  at  10^  ? 

6.  What  is  the  amount  of  $3,750  for  3  years,  at  h%  ? 

7.  February  11,  1880,  I  borrowed  $2,250.     How  much 
did  the  debt  amount  to,  February  11, 1883,  interest  at  6^? 

8.  What  is  the  interest  of  $560.10  for  2  years,  at  1%% 

Case  II.  Interest  for  niontlis. 

Oral  TTo^A'.— 336.  A.  1.  What  part  of  a  year  is  1 

month  ? 


2.  Are  6  mo.  ? 

3.  Are  3  mo.  ? 


Jf..  Are  4  mo.? 
5.  Are  2  mo.? 


6.  Are  9  mo.  ? 

7.  Are  8  mo.  ? 


8.  Are  5  mo.  ? 

9.  Are  7  mo.  ? 


J?.  1.  The  per  cent  on  $1  for  1  month  is  what  part  of 
the  rate  on  $1  for  1  year? 


PERCENT  A  OE.— INTEREST. 


259 


2.  At  %%  per  annum,wliat  is  the  per  cent  on  $1  for  6  mo.  'i 


S.^ov  3  mo.? 
Jf..  For  9  mo.  ? 


5.  For  4  mo.  ? 

6,  For  8  mo.  ? 


7.  For  2  rao.  ? 

8,  For  1  mo.  ? 


9.  For  7  mo.  ? 
10,  For  11  mo  J 


(7.  i.  The  interest  of  $1  for  1  month  is  what  part  of 
the  interest  of  $1  for  1  year  ? 

^.  The  interest  of  any  sum  for  1  month  is  what  part 
of  the  interest  of  the  same  sum  for  1  year  ? 

3.  At  6^  per  annum,  what  is  the  interest  of  $1 

6.  For  9  mo.  ?     6".  For  8  mo.  ?     10.  For  7  mo.  ? 

7.  For  4  mo.  ?     9.  For  2  mo.  ?     ii.  For  11  mo.  ? 

%  per  annum,  what  is  the  interest  of 


Jf.  For  1  mo.  ? 
^.  For  3  mo.  ? 


jD.  At 


1.  $4  for  3  mo.  ? 
^.  $2  for  9  mo.  ? 
^.  $7  for  4  mo.  ? 
4.  $10  for  8  mo.? 


5.  $20  for  2  mo.  ? 
^.  $100  for  1  mo.  ? 
7.  $40  for  7  mo.  ? 
5'.  $12  for  11  mo.? 


9.  $1  for  1  yr.  6  mo.  ? 
i<9.  $8  for  1  yr.  6  mo.  ? 
ii.  $40  for  1  yr.  4  mo.  ? 
12.  $200  for  3  yr.  7  mo.  ? 


Process. 

$8Jf.60 

.06 


397.       j  III.  /?z^.  for  1  mo.  =  7^  of  int.  for  1  yr. 
Formulas.  \  IV.  Int.  for  mo.  =  j%-  of  int.  for  1  yr.  x  No.  of  mo. 

Written  Work.—E:s..  What  is  the  interest  of  $84.50 
for  1  yr.  5  mo.,  at  Q%  ? 

Explanation.— Is^.  Forlyr.  I  multi- 
ply $84.50,  the  principal,  by  .06,  the 
rate  of  interest,  and  obtain  $5.07. 

2d.  For  1  yr.  5  mo.,  or  11  mo.  $5,0  70     lut. forlyr. 
Since  17  months  are  ^  years,  ^  ^ 

I  multiply  $5.07,  the  interest  o  o    in — 
for  1  year,  by  ^,  and  obtain      2id)$00.1t^ 

$7. 18,  the  required  interest.  $7.18  Int.  forlyr.  5  mo. 

a.  When  the  time  is  expressed  in  months : — Divide  the  in- 
terest for  1  year  by  12,  and  multiply  the  quotient  hy  the 
number  expressing  the  time  in  months. 

h.  Or,  to  avoid  fractions  in  the  process : — Multiply  the  in- 
terest for  1  year  by  the  number  expressing  the  time  in 
months,  and  divide  the  product  by  12. 


260 


SECOND   HOOK  IN  ARITHMETIC. 


Problems. 
L  What  is  the  interest  of  $952.17  for  3  months,  at  6^? 
^.  What  is  the  interest  of  $187.75  for  5  months,  at  4;^? 

5.  Find  the  interest  of  $1,168.48  for  1  yr.  4  mo.,  at  Z\%, 
^.  If  I  have  $938.25  on  interest  for  4  yr.  10  mo.,  at  6^, 

how  much  interest  shall  I  receive  ? 

J.  What  is  the  amount  of  $294.25  for  6  mo.,  at  5^? 

6.  If  I  give  my  note  for  $3,275,  Jan.  11, 1882,  and  pay 
it  Feb.  11, 1883,  with  ^%  interest,  how  much  do  I  pay  ? 

7.  If  I  borrow  $2,732,  at  12^  interest,  and  pay  the  debt 
in  1  month,  how  much  do  I  pay  ? 

8.  How  much  must   I  pay   for  the    rent   of  a  store 
valued  at  $3,150,  for  1  yr.  4  mo.,  at  7^  on  the  valuation  ? 

9.  How  much   is   the   semi-annual   interest   on  a  7^ 
mortgage  for  $1,730? 

Case  III.   Interest  for  days. 
Oral  Work.— 32^. 


A.*  1.  IIow  many  days  is 

.1  of  a  mo.  ? 
2.  Are  .3  of  a  mo.  ? 


How  many  tenths  of  a  month 
9,  Is  1  day?    |    10,  Are  3  days? 


11,  15  da.  ?  15,  24  da.  ?  19,  25  da.  ? 

3.  .2  mo.  ?     6.  .9  mo.  ?     12,  9  da.  ?  16,  \%  da.  ?  20,  1  da.  ? 

Jf,  .6  mo.  ?     7.  .0^  mo.  ?   i^.  6  da.  ?  i7.  11  da.  ?  21,  16  da.? 

5.  .4  mo.  ?     6*.  .5|  mo.  ?   i^.  12  da.  ?  18.  20  da.  ?  ^^.  29  da.? 

JB.  1,  The  per  cent  on  $1  for  3  days  is  what  part  of 
the  per  cent  on  $1  for  1  month  or  30  days  ? 

2,  At  %%  per  annum,  what  is  the  %  on  $1  for  3  days  ? 


S.  For  15  da.^ 
4.  For  9  da.  ? 


5,  For  12  da.? 

6,  For  18  da.? 


7.  For  1  da.  ? 
5*.  For  11  da.? 


9,  For  20  da.  ? 
m  For  25  da.  ? 


O.  i.  The  interest  of  $1  for  3  days  is  what  part  of  the 
interest  of  $1  for  1  month  ? 

2.  The  interest  of  any  sum  for  3  days  is  what  part  of 
the  interest  of  the  same  sum  for  1  month  % 


PER  CENT  A  GE.— INTEREST. 


261 


At  Q%  per  annum,  what  is  the  interest  of  $1 


S.  For  1  mo.  ? 
^.  For  15  da.? 


5.  For  24  da.  ? 
(5.  For  3  da.  ? 


7.  For  1  da.  ? 
5'.  For  4  da.  ? 


P.  For  20  da.  ? 
m  For  25  da.  ? 


It  Of  $6  for  12  da.  ?  I  12.  $25  for  18  da.  ?  |  i^.  $100  for  27  da.  ? 
1>.  How  many  months  and  tenths  of  a  month 


«^.  Arel  mo.  21  da.? 
4.  Are  10  mo.  6  da.? 


^.  Are2  mo.  15  da.? 
^.  Are  6  mo.  18  da.? 

7.  Are  1  yr.  3  mo.  12  da.? 

8.  Are  2  yr.  1  mo.  27  da.  ? 

M.  At  6^  per  annum,  what  is  the  interest  of  $1 


5.  Are  7  mo.  25  da.  ? 
^.  Are  11  mo.  29  da.? 
9.  Are  1  yr.  8  mo.  13  da.? 
10.  Are  3  vr.  28  da.? 


i.  For  2  mo.  15  da.? 
2.  For  3  mo.  3  da.  ? 

7.  Of 

8.  Of 


5.  For  7  mo.  25  da.? 
^.  For  5  mo.  17  da.? 


.^.  For  1  mo.  21  da.? 

Jf.  For  4  mo.  10  da.? 
)  for  6  mo.  3  da.  ?    1 10.  Of  $1  for  1  yr.  3  mo.  12  da.  ? 
for  10  mo.  18  da.  ?  1 11.  Of  $10  for  1  yr.  2  mo.  6  da.  ? 


9.  Of  $9  for  7  mo.  17  da.  ?   1  i^.  Of  $5  for  4  yr.  4  mo.  20  da.  ? 
At  any  rate  of  interest,  how  is  the  interest  of  any  prin- 
cipal found 
13.  For  3  days?  |  U.  For  1  day  ?  |  15.  For  any  number  of  days? 

3^0.     j    V.  Int.  for  3  da.  =  .1  of  int.  for  1  mo. 
Formulas.  (  VI.  Int.  for  da.  nr  .1  of  int.  for  1  mo.  x  i  the  No.  of  da. 

Written  Work.— Ex.  What  is  the  interest  of  $169.75 
for  2  yr.  3  mo.  10  da.,  at  8^? 

Second  Process. 
2  yr.  3  mo.  10  da.  =  2  7.3^  mo. 

$169.75    Prin. 

.0  8    Rate  of  int. 
$13.5800    Int.forlyr. 

2  7.3^- 
12)$37 1.187 

$30,932    Iiit.for?Ii^yr. 


First  Process. 

2  yr..  3  mo.  10  da.  —  27.3^  mo. 

$16  9.75    Prin. 

.  0  8   Rate  of  int. 
12)$13.5800    Int.forlyr. 
$1.13^        Int.  fori  mo. 
2  7.3^        No.  of  mo. 

Required  Int. 


$30.93'. 
The  second  process  avoids  complicated  computations  in  fractions. 


262  SECOND   BOOK  IN  ARITHMETIC. 

Problems. 
.  1.  How  mucli  interest  will  I  have  to  pay,  at  7^,  on  a 
loan  of  $1,296  for  9  mo.  15  da.  ? 

^.  What  is  the  interest  of  $716.25  for  1  yr.  24  da.,  at  6^? 
3,  Find  the  interest  of  $936  for  3  yr.  2  mo.  29  da.,  at  10%, 
^.  What  is  the  interest  of  $718  for  1  yr.  14  da.,  at  6^? 

5.  If  I  borrow  $819  for  20  days,  at  8^,  how  much  inter- 
est must  I  pay  ? 

6,  What  is  the  interest  of  $483.70,  from  Nov.  23, 1882, 
toDec.8, 1883,  at  7^? 

330.  Rules  foe  Interest. 

I.  To  find  the  interest  for  1  year:— 
Multijplij  the  jprincijpal  hy  the  rate  of  interest 

II.  To  find  the  interest  for  2  or  more  years  :— 
Multiply  the  interest  for  1  year  hy  the  number  of  years. 

III.  To  find  the  interest  for  any  other  time  :— 

MuUijply  the  interest  for  1  year  hy  the  time  expressed 
in  months  and  tenths  of  a  months  and  divide  the  product 
hy  m, 

IV.  To  find  the  amount:— 
Add  the  interest  to  the  principal. 

a.  In  computations,  carry  the  partial  results  to  four  decimal  places. 

b.  In  final  results,  if  the  mills  are  5  or  more,  call  them  1  cent; 
and  if  they  are  less  than  5,  reject  them. 

331.  All  computations  of  interest  come  under  this 

General  Formula. 
Principal  X  rate  of  interest  x  time  =  interest. 


PER  CEN  TA  G  E.—IN  TER  ES  T. 


263 


'for 


5  yr., 

2  yr.  9  mo., 

I  yr.  7  mo.  21  da., 

II  mo.  13  da., 

3  yr.  28  da.. 


at 


10^? 

mi 
li^per  mo.  ? 


atj  7^? 


Peoblems. 
What  is  the  interest 

1-  25.  Of  $387.50 
26-  60.  Of  1293 
61-  76.  Of  $7,461.13 
76-100.  Of  $12,009.08 
101-126.  Of  $4,731.87  ^ 

What  is  the  amount 
126-13Jf..  Of  $82.44    1  f  7  mo.  10  da., 

136-US.  Of  $316.90  \^ov\  2  yr.  11  mo., 
lJi.Ji.-162.  Of  $2,054    J  U  yr.  3  mo.  27  da.,, 

163.  A  contractor,  while  building  a  church,  borrowed 
$13,080  for  Y  mo.  15  da.,  at  %.  How  much  interest  did 
he  pay? 

161i..  Find  the  amount  of  $180  for  2  yr.  2  mo.  20  da.,  at  6^. 

165.  At  %%  a  month,  how  much  will  a  banker  receive 
for  the  use  of  $100  for  2  yr.  5  mo.  10  da.  ? 

156.  A  mortgage  for  $375  has  been  running  4  yr.  4 
mo.  15  da.     How  much  interest  has  accrued  on  it,  at  8^? 

157.  What  is  the  interest  of  $873.60,  from  Feb.  10, 
1881,  to  Oct.  10,  1882,  at  10^? 

158.  A  note  for  $1,824.75,  dated  Oct.  8,  1881,  was  paid 
Dec.  23, 1882,  with  %%  interest.    What  amount  was  paid  ? 

159.  /i'/J'/Z^-.  New  York,  JwZ2/ 1, 18S2. 


This  note  was  paid  Nov.  17,  1882,  with  7^  interest. 
What  was  the  amount? 

160.  What  is  the  discount  on  a  bill  of  goods  amounting 
to  $237.50,  at  30  days,  2^^  off  for  cash? 


264  SECOND    BOOK  IN  ARITHMETIC. 

161,  Draw  your  note  for  $956,  bearing  date  Boston, 
Oct.  9,  1882,  payable  to  James  Fields,  or  bearer,  and  due 
June  15,  1883,  with  interest. 

16^,  Compute  the  interest  on  this  note,  at  6^. 

163,  Draw  your  note  for  $250,  bearing  date  Chicago, 
Jan.  14, 1882,  payable  to  Thomas  Clark,  or  order,  and  due 
in  1  year,  with  10%  interest  after  3  months. 

Find  the  amount  due  to-day  on  this  note. 

16Jt„  A  debt  of  $1,000  due  Mar.  2, 1881,  was  paid  Apr. 
17, 1882,  with  interest  at  S%,    What  was  the  amount  ? 

165,  At  the  date  last  named,  a  payment  of  $350  was 
made.     How  much  was  due  June  26, 1883  'f 

166,  Memorandum :— Note  for  $1,824,  dated  Philadel- 
phia, Oct.  10, 1881.  Payment  made  of  $550,  Apr.  25, 1882. 
What  was  due  Jan.  2, 1883,  interest  at  6^? 

167,  ^'f  .650.  St.  Louis,  Mo.,  A'ov.  19, 18S0. 


oic/et,   o/c^x^een    c^unc^iec^  ^c/iCy   ^ouata,  lotm  tnieted^,   /oi  va/ue 

Indorsements  :—Z\m^  17,1882,  $225;  Oct.  24,  1882, 
$475.  How  much  was  due  on  settlement,  Mar.  3,  1883, 
interest  at  6^? 

168,  A  wholesale  merchant  sold  a  bill  of  goods  amount- 
ing to  $3,1263^^0-,  on  a  credit  of  3  months;  and  for  cash 
down,  he  deducted  or  discounted  5%  from  the  amount  of 
the  bill.     How  much  did  he  receive  for  the  goods  ? 

169.  An  invoice  of  fancy  goods,  at  retail  prices,  amounts 
to  $920 ;  and  the  discounts  are  25^  off  from  the  amount 
of  the  invoice,  and  Sfo  off  from  the  balance,  for  cash. 
What  is  the  cash  cost  of  the  goods? 

For  outlines  of  percentage  for  reyiew,  see  page  275. 


PER  CENT  A  GE.—RE  VIE  W.  265 

General  Eeview  Peoblems  in  Percentage. 

1,  At  an  election,  the  successful  candidate  received 
11,480  votes,  and  the  defeated  candidate  87|-^  of  the  same 
number.     How  many  votes  did  both  candidates  receive  ? 

^.  A  farmer  divided  225  acres  of  land  among  his  3 
sons,  giving  81  acres  to  the  first,  Y6.5  acres  to  the  second, 
and  the  remainder  to  the  third.  What  %  of  the  whole 
number  of  acres  did  each  receive  ? 

3,  What  is  the  difference  between  33^  and  25|-^  of 
480  miles  ? 

^.  A  merchant  sold  goods  at  33^^  above  cost,  and  re- 
ceived $  ,45  a  yard  for  them.  How  much  did  they  cost 
him  per  yard  ? 

5.  I  sold  2  stones  for  $24  apiece ;  on  one  of  them  I 
made  20^,  and  on  the  other  I  lost  20^.  What  was  my 
gain  or  loss  on  both  ? 

6.  If  10^  is  lost  by  selling  boards  at  $7.20  per  M, 
what  %  would  be  gained  by  selling  them  at  $  .90  per  C  ? 

7.  At  what  price  per  pound  must  I  sell  tea  that  cost 
*  $  .75,  to  make  a  profit  of  20^  ? 

8.  A  commission-merchant  received  $157.75  for  sell- 
ing flour,  his  commission  being  2|-  per  cent.  For  how 
much  was  the  flour  sold  ? 

9.  A  lawyer  collected  $950,  and  charged  1^^%  com- 
mission. What  were  his  fees,  and  what  was  the  sum  to 
be  remitted  ? 

10,  If  a  tax  of  $387.75  is  paid  by  an  assessment  of  f 
of  1^,  what  is  the  assessed  valuation  ? 

11.  The  cost  of  building  a  school-house  is  $935 ;  the 
assessed  valuation  of  the  district  is  $34,750,  and  A  owns 
a  farm  assessed  at  $9,475.     How  much  is  his  tax  ? 

M 


266  SECOND    BOOK  IN  ARITHMETIC. 

12,  A  merchant's  store  is  insured  for  $7,850,  at  \% ;  and 
his  goods  for  $12,375,  at  \%,   What  premium  does  he  pay  ? 

IS,  A  steamship  was  insured  for  $87,500,  the  premium 
being  $1,968.75.     What  was  the  per  cent  ? 

lit,.  If  I  sell  railroad  stock  which  cost  me  $2,500,  at  a 
loss  of  8 J^,  how  much  do  I  receive  for  it  ? 

15,  How  much  are  5  shares  of  bank  stock  worth  at  118  ? 

16,  An  account  of  $45.50  has  been  due  2  yr.  6  mo. 
How  much  interest  has  accrued  on  it,  at  6^  ? 

17,  What  is  the  interest  of  $735  from  April  9,  1882, 
to  July  15, 1884,  at  6  per  cent  ? 

18,  How  much  will  a  debt  of  $385.50,  contracted  Feb. 
15, 1882,  amount  to  Jan.  3, 1884,  on  interest  at  1%\ 

19,  If  you  save  $150  of  your  salary  each  year,  and  put 
your  savings  at  interest,  at  10^,  at  the  end  of  each  year, 
how  much  will  your  savings  amount  to  in  10  years  ? 

W,  After  spending  25;^^  of  my  capital,  and  25^  of  the 
remainder,  I  had  $675.     What  capital  had  I  at  first  ? 

21,  A  merchant  bought  cassimere,  at  auction,  at  28^^ 
below  the  manufacturer's  price,  paying  $1.25  a  yard  for 
it.  He  retailed  it  at  22^  above  manufacturer's  price.  At 
what  price  did  he  retail  it  ? 

22,  I  sell  coal  at  the  same  price  per  net  ton  (2,000  lb.) 
as  I  pay  for  it  per  gross  ton  (2,240  lb.).  What  per  cent 
profit  do  I  make? 

23,  A  market -man  sells  eggs  at  J  cent  apiece  above 
cost,  and  makes  25^.  What  do  the  eggs  cost  him  apiece  ? 
At  what  price  per  dozen  does  he  sell  them  ? 

2Jf..  The  premiums  paid  for  insuring  four  stores  in  a 
block  are  $147,  $97.50,  $153.75,  and  $107.25 ;  and  the 
rate  is  1^%.    What  amount  is  insured  on  each  store  ? 


PER  CENT  A  GE.—RE  VIE  W.  267 

25.  A  commission-mercliaiit  in  Milwaukee  received  from 
an  Oswego  miller  $1,000,  to  buy  wheat,  after  deducting 
his  commission  at  2%.  The  merchant  paid  $1.06^  per 
bushel.     How  many  bushels  did  he  buy? 

26.  If  I  invest  $2,500  in  bank-stock,  and  sell  it  at  an 
advance  of  6^,  for  how  much  do  I  sell  it  ? 

27.  I  buy  stock  at  5  per  cent  discount,  and  sell  it  at  3 
per  cent  premium,  gaining  $180.  How  much  did  I  invest  ? 

28.  Dec.  15,  1877,  a  farmer  mortgaged  his  farm  for 
$4,850  ;  and  Sept.  15,  1881,  he  paid  the  mortgage,  with 
6^  interest.     What  amount  did  he  pay  ? 

29.  A  note  for  $754.19,  dated  Jan.  10,  1881,  was  paid 
Dec.  14, 1882,  with  6^  interest.    What  amount  was  paid  ? 

30.  What  sum  must  be  paid  to  cancel  a  debt  of  $219.16, 
which  has  been  due  1  yr.  6  mo.  14  da.,  at  the  legal  rate  of 
interest  in  this  State  ? 

31.  A  man  who  owned  f  of  a  ship,  sold  40^  of  his  share. 
What  part  of  the  ship  did  he  then  own  ? 

32.  A  grain  dealer  bought  wheat  at  $1.25  per  bushel, 
and  sold  it  at  a  profit  of  20^,  making  $50  by  the  transac- 
tion.    How  many  bushels  did  he  buy? 

33.  A  merchant  sells  goods  at  retail  at  Z0%  above  cost, 
and  at  wholesale  at  12^  less  than  retail  price.  What  is 
his  gain  per  cent  on  goods  sold  at  wholesale  ? 

J^.  A  man  27  years  of  age  took  out  a  life-insurance 
policy  for  $8,000  for  the  benefit  of  his  wife,  at  the  annual 
rate  of  $21.70  per  $1,000 ;  his  death  occurred  at  the  age 
of  f33.  How  much  did  the  widow  receive  more  than  had 
been  paid  in  annual  premiums? 

35,  I  sold  375  100-pound  sacks  of  Kio  coffee,  at  $  .17^ 
a  pound,  and  my  commission  amounted  to  $80.27.  What 
per  cent  commission  did  I  receive  ? 


268  SECOND    BOOK   IN   ARITHMETIC. 

36.  A  man  sold  his  liouse  for  $2,500,  payable  in  one 
year,  with  interest  at  Q%.  At  the  end  of  6  months  he 
received  a  payment  of  $1,600.  Wliat  was  due  him  at  the 
end  of  the  year  ? 

37.  How  much  will  you  gain,  if  you  buy  45  shares  of 
telegraph  stock  at  27^  discount — or  27%  below  par — and 
sell  it  at  12^  discount  ? 

38.  If  goods  are  bought  at  J  of  their  value,  and  sold  for 
10%  more  than  their  value,  what  is  the  gain  per  cent  ? 

39.  A  merchant  bought  a  hogshead  of  molasses,  and  lost 
J  of  it  by  leakage ;  he  sold  the  remainder  at  20^  advance 
on  its  cost.    What  per  cent  did  he  lose  on  the  investment  ? 

4,0.  A  note  for  $850,  at  6^  interest,  was  given  Sept.  15, 
1880.  April  23,  1881,  a  payment  of  $290  was  made ;  and 
Jan.  3,  1882,  a  payment  of  $345.  How  much  was  due 
Sept.  29,  1882  ? 

Ji,l.  A  druggist  buys  perfumery  at  -J-  off  from  retail  price. 
What  %o  does  he  make  by  selling  it  at  the  retail  price  ? 

^.  A  man  can  hire  a  farm  of  97  acres  for  $500  per 
annum,  or  he  can  buy  it  for  $70  an  acre.  If  money  is 
worth  6^,  which  is  the  cheaper  for  him,  and  how  much 
the  cheaper? 

J^S.  A  farmer  bought  80  sheep,  at  $4.20  a  head,  giving 
his  note  payable  in  6  months,  at  7^.  At  the  end  of  the 
6  months  he  sold  the  sheep  at  $5.25  a  head,  and  paid  the 
note.    How  much  did  he  get  for  the  keeping  of  the  sheep  ? 

^^.  If  wool  that  costs  $  .60  per  pound,  shrinks  45^  in 
cleansing,  at  what  price  per  pound  must  it  be  sold,  to  gain 
33^^  on  the  cost  ? 


BLACKBOARD  OUTLINES. 
FOR   REVIEWS   AND    EXAMINATIONS. 


Notation  and  Numeration. 


I.  Terms. 


1.  Arithmetic, 

2.  A  unit. 

3.  A  numher. 

4.  Unit  of  a  number. 

5.  An  integer. 

Whole  numbers. 

6.  Cipher  or  zero. 
Y.  Digits, 

.  8.  Periods  of  figures. 


9.  Order  of  a  unit. 

10.  Values  expressed  hy  figures. 

11.  Simple  value. 

12.  Local  value. 

13.  Notation. 

14.  Numeration, 

15.  Dollar  m,arlc. 

16.  Decimal  point. 


TT   T>  O-  Fa^wes  of  orders  from  right  to  left. 

II.  Principles,  i  ^   Tr  ;       ^     j      x        i  j^*  *      •  z.^ 

(  2.  Values  of  orders  from  left  to  right. 

fC  a.  Dollar  mark. 
1.  For  expressing  money.  <  b.  Decimal  point. 
Vc   Less  than  10  cents. 
2.  For  notation. 
3.  For  numeration. 


I.  Terms. 


'  1,  Like  numbers. 

2.  Unlike  numbers. 

3.  Addition. 

4.  Parts. 

~  5.  /Swm  or  amount. 


Addition, 

II.  Signs. 

III.  Principles.  \ 

IV.  Rule. 


j  1.  Of  addition, 
i  2.  Of  equality. 

1.  Pa7'is  =  the  whole. 

2.  What  units  can  be  added. 
I,  II,  III. 


I.  Terms. 


1.  Subtraction. 

2.  Minuend. 

3.  Subtrahend. 

4.  Difference  or  remainder 


Subtraction. 

II.  Sign. 


III.  Principle. 

IV.  Rule.  Steps  I,  II,  III. 


270 


SECOND   BOOK  IN  ARITHMETIC. 


I.  Terms.  - 


II.  Sign. 


Multiplication. 

1.  Multiplication. 

Multiplication  is  addition. 

2.  Multiplicand. 
8.  Multipliei'. 

4.  Factors. 


5.  Product. 

Partial  products. 

6.  A  concrete  number. 
Y.  An  abstract  numbtr. 


III.  Important  Facts. 


IV.  Principles. 


'  1.  Multiplier  an  abstract  numh€i\ 

2.  Either  factor  for  midtipliei\ 

3.  Annexing  0  to  a  numbei'. 
L  4.  One  factor  0. 

'  1.  Product  concrete  or  abstract. 

2.  Removing  a  number  to  tlie  left. 

3.  Multiplying  by  ones^  tenSj  hundredsy  etc. 
.  4.  Number  of  0^8  on  right  of  product. 


y.  Cases  and  Rules. 


'  I.  77i€  multiplier  a  digit. 

2.  Ciphers  on  the  right  of  multiplier. 

3.  The  multiplier  two  or  more  digits. 

.  4.  Ciphers  on  the  right  of  both  factors. 


I.  Terms. 


Division. 

1.  To  divide. 

2.  I&actional  parts. 

Halves,  thirds,  and  so  on. 

3.  Division. 

Division  is  subtraction. 

4.  Long  division. 


5.  Short  division. 

6.  Dividend. 

Partial  dividend. 

7.  Divisor. 

8.  Quotient. 

9.  Average  of  numbers. 


II.  Signs.    1st;  2d. 


III.  Important  Facts. 


IV.  Principles, 


>i 


1.  Divisor  an  abstract  number. 

2.  Steps  in  division. 

3.  How  to  change  dollars  and  cents  to  cents. 

4.  Wlien  to  change  dividend  to  cents. 
^  5.  How  to  divide  dollars  by  10  or  100. 

1.  Quotient  like  dividend. 

2.  Removing  a  number  to  the  right. 


V.  Cases  and  Bulks.  \  ^-  ^I'T''  ««'%»«''«  ^i*- 
(  2.   Ciphers  on  right  of  divisor. 


BLACKBOARD    OUTLINES. 


271 


Decimals. 


X.  Terms. 


1.  A  decimal  unit. 

2.  A  decimal. 

3.  A  mixed  member. 

4.  The  decimal  point. 

5.  Currency. 

^  6.  Money  units. 


Y.  Reduction. 

8.  A  debt. 

9.  A  debtor. 
10.  A  creditor. 
\l.  A  billofgoo4s. 
12.  An  item. 


13.  Extending  an  item. 

14.  The  footing  of  a  bill 

15.  ^4n  account. 

16.  The  balance  of 

an  account. 

1 7.  Receipting  a  bill. 


11.  Important  Facts. 


IIL  Principles. 


IV.  Processes 
AND  Rules. 


'  1.  How  to  write  cents  and  mills. 

2.  How  to  read  cents  and  mills. 

3.  Removal  of  decimal  point  to  the  Hght. 
.  4.  Removal  of  decimal  point  to  the  left. 


(I.  Notation, 


\: 


2.  Reduction. 


2d.  Removing  decimal  ciphers. 


1st.  Lower  to  higher  units. 
d.  Higher  to  lower  units, 
j  1st.  Annexing  decimal  ciphers. 

3.  Addition. 

4.  Subtraction. 

5.  Multiplication.    Number  of  decimal  places  in  product. 
1 6.  Division.    Number  of  decimal  places  in  quotient. 

1.  To  reduce  decimal  currency, — 8  processes. 

2.  How  to  add  decimal  numbers. 

3.  How  to  subtract  decimal  numbers. 

4.  Ride  for  multiplication. 

Less  decimal  places  in  product  than  in  factors. 

6.  Rule  for  division. 

When  annex  decimal  ciphers  to  dividend. 


Properties  of  Numbers. 


I.  Terms. 


1. 

Integral  factors. 

G. 

Factoring. 

An  integer  is  exactly 

1. 

An  exact  divisor. 

divisible. 

8. 

A  common  divisor. 

2. 

A  composite  number. 

9. 

The  greatest  common  divisor. 

3. 

A  prime  number. 

10. 

A  multiple. 

A  prime  factor. 

11. 

A  common  multiple. 

4. 

An  even  numbe7\ 

12. 

The  least  common  multiple. 

6. 

An  odd  number. 

13. 

Cancellation. 

272 


SECOND    BOOK  IN  ARITHMETIC. 


II.  Principles. 


III.  Rules. 


1.  The  greatest  commo?i  divisor  the  product  of  what? 

2.  The  least  common  multiple  a  multiple  of  wliat  ? 

3.  Cancelling  a  factor  from  a  number. 

4.  Cancelling  a  common  factor. 

When  the  factor  1  remains. 


1 .  For  finding  prime  factor's. 

2.  For  finding  greatest  common  divisor. 

3.  For  finding  midtiples;  1,  S. 


I.  Terms. 


Fractions, 

1. 

2. 

Fractional  parts. 
A  fractional  unit. 

10. 

A  compound  fraction^. 
The  word  of 

3. 
4. 

5. 
6. 

A  fraction. 
TJie  terms. 

Lowest  terms. 
The  denominator. 
The  numerator. 

11. 
12. 
13. 
14. 
15. 

A  complex  fraction. 
The  unit  of  a  fraction. 
To  analyze  a  fraction. 
Similar  fractions. 
Least  similar  fractions. 

7. 
8. 
9. 

A  proper  fraction. 
An  improper  fraction. 
A  mixed  number. 

16. 
17. 
18. 

Dissimilar  fractions. 
Common  denominator. 
Least  common  deiiominator. 

Decimal  fractions. 

19. 

A  fraction  is  inve^'ted. 

IT    Important  Fapt's   ^  ^'  ^^^^^'^  <^^^^^^^^l(>^  ^  ^^^^^^^^''^  ^^^l^P^^- 

*  (  2.  Least  common  denominator  least  common  multiple. 


III.  Principles. 


IV.  Cases 
AND  Rules. 


rl.  Dividing  terms  by  a  common  factor. 

I  2.  Multiplying  terms  by  any  number. 

I  3.  What  the  product  of  two  fractions  equals. 

L  4.  Tlie  process  of  dividing  by  a  fraction. 

"  1.  Fractions  to  lowest  terms. 

2.  Fractions  to  given  fractional  units. 

3.  Dissimilar  to  similar  fractions. 

4.  Dissimilar  to  least  similar  fractions. 

5.  Fractions  to  integers  or  mixed  numbers. 
C.  Integers  or  mixed  numbers  to  improper 

fractions. 
Use  of  each  case  in  reduction. 
Steps  1,  2. 
Steps  1,  2. 
Steps  1,  2. 
Steps  1,  2. 


Reductions  of 


7.  Addition. 

8.  SvI)traction 

9.  Multiplication, 
'-10.  Division. 


BLACKBOARD    OUTLINES. 


273 


Compound  Numbers. 


I.  Terms. 


'1,  A  simple  number. 

2.  A  denominate  number. 

3.  A  compound  number. 

4.  A  surface. 

5.  Area. 

Square  inch  ;  foot ;  yard. 

6.  A  solid  or  body. 

Cubic  inch ;  foot ;  yard. 
^1.  Weight 


II.  Tables  of  Measures. 


III.  Important  Facts.  - 


8.  Linear  measures. 

9.  Surface  measui'es. 

10.  Solid  measures. 

1 1 .  Liquid  measures. 

12.  Dry  measures. 

13.  Avoirdupois  weights. 

14.  Time. 

15.  Reduction  descending. 

16.  Reduction  ascending. 

5.  Dry. 

6.  Avoirdiipois  weight. 
1.  Time. 
8.  Counting. 


1.  Linear  or  line. 

2.  Surface. 

3.  Solid. 
L4.  Liquid. 

1 .  ^010  quantities  of  articles  are  determined. 
.  2.  Process  used  iji  reduction  descending. 

3.  Process  used  in  reduction  ascending. 

4.  Special  7'ules  omitted,  and  why. 

5.  Leap-years,  how  deter mhied. 

6.  A  month,  in  business  transactions. 
VL 


lY.  Cases.    /;  //;  ///;  IV;  V; 

'1.  Rule  for  reduction  descending. 

2.  Rule  for  reduction  ascending. 

3.  How  to  add,  subtract,  multiply,  and  divide 
compound  numbers. 

4.  How  to  divide  by  a  compound  number. 
^  5.  How  to  find  difference  in  time  between  datesi 


V.  Processes  and  Rules. 


Measurements. 


I  Terms.  ^ 


1.  Dimensions. 

2.  A  line. 

3.  A  surface. 

4.  A  body. 

5.  Extension. 

6.  An  angle. 

7.  A  right  anpte. 

8.  An  acute  angle. 

9.  An  obtuse  angle. 

10.  Perpendlcidar  line 

11.  A  parallelogram. 
^11.  A  rectangle. 


13.  ^  square. 

14.  Altitude  of 

parallelogram. 

15.  Diagonal  of. 

16.  Perhnete)'  of. 
Vl.  A  triangle. 

18.  ^  right-angled. 

19.  -4w  obtuse-angled. 

20.  ^?i  acute-angled. 

21.  ^ase  o/  triangle. 

22.  Fi'j'^^'x  o/. 

23.  Altitude  of. 

M2 


24.  -4  trapezoid. 

25.  Altitude  of. 

26.  ^  drc/«?. 

27.  Circumfei'ence  of. 

28.  ^rc  o/. 

29.  Diameter  of. 

30.  Radius  of. 

31.  ^  rectangular 

solid. 
2>2.A  cube. 
33.  (7w62c  contents  or 

volume. 


274 


SECOND    BOOK  IN  ARITHMETIC, 


2.  F<yr  measures  of  surf  ace. 


II.  Units,  -j  3.  For  measures  of  vobtme. 


'  1.  Standard  unitSy  table  of. 

f  a.  The  square  foot. 
A  board  foot. 

b.  The  square  yard. 

c.  The  square. 
.  d.  A  bunch  of  lath. 

a.  The  cubic  foot. 
A  timber  foot. 

b.  The  cubic  yard. 

c.  The  barrel. 

d.  The  hogshead. 

e.  The  perch. 
/.  A  brick. 

fa.  A  bushel. 
I        Stricken  measure. 
I         Heaped  measure. 
[  6.  A  ton  of  hay. 

6.  Allowances  or  deductions.  ]  .  *  t  ^^ 


4.  For  farm  products. 


III.  Important  Facts.  - 


1.  Units  in  area  of  a  rectangle. 

2.  Units  in  which  dimensions  are  always  expressed, 

3.  Dimensions  must  have  a  common  unit. 

4.  Units  in  either  dimension  of  a  rectangle. 

5.  Units  in  area  of  a  parallelogram. 

6.  Units  in  area  of  a  trapezoid. 
*J.  Units  in  area  of  a  circle. 

8.  Units  in  volume  of  a  rectangidar  solid. 
.  9.  Units  in  one  dimension  of  a  rectangular  solid. 


IV.  Cases 
AND  Rules. 


1.  Rectangles 
and  triangles. 


2.  The  circle. 


a.  Area  of  a  rectangle. 

b.  Either  dimension  of  a  rectangle. 

c.  Area,  base,  and  altitude  of  a  triangle. 

d.  Area  of  a  parallelogram. 
€.  Area  of  a  trapezoid. 

a.  Circumference  or  diameter. 
For  great  accuracy. 

b.  Area. 
^    ^  ,        7        ,.,(«.  Cubic  contents  or  volume. 

_  3.  Kectangulur  eohds.  |  j  ^^^  din^ension. 


BLACKBOARD   OUTLINES. 


275 


Percentage, 

A.  Without  Time  as  an  Element. 


I.  Terms. 


1.  Per  cent. 

2.  Percentage. 

3.  The  per  cent. 

4.  The  base. 

5.  The  percent- 

age. 

6.  The  amount. 

7.  The  difference. 

8.  Profit. 

9.  Loss. 

10.  An  agent. 
W.  A   commission- 
merchant. 

12.  Commission. 

13.  Insurance. 

14.  Valuation. 

15.  Premium. 

16.  A  policy. 
IT.  Taa^es. 

18.  -4  property  tax. 
^19.  A  poll  tax. 


20.  Customs  or  duties. 

21.  Imports. 

22.  A  custom-house, 

23.  ^  jtjorif  o/  cn^r?/. 

24.  ^  ^anl/f. 

25.  ^c?  valorem  duties. 

26.  Specific  duties. 

27.  :7'are. 

28.  Leakage. 

29.  Breakage. 

30.  6^ross  weight. 

31.  iV^^  weight. 

32.  ^n  invoice. 
33.-4  manifest. 
34.  /S/oc;^-. 
35.-4  sAare. 

36.  Par  ?;aZwc. 

37.  Market  value. 

At  par. 
Above  par. 
Below  par. 


38.  Premium. 

39.  Discount. 

40.  -4  stock-broker. 

41.  Brokerage. 

42.  A  partnership. 

43.  -4  ^rm  or  /i02/.s<?. 

44.  Partners. 

Active. 
Silent. 
General. 
Special. 

45.  Capital  or  stock, 

46.  Resources. 

47.  Liabilities. 

48.  iVe^  capital. 

49.  ^  <7e/c/^. 

Solvency. 
Insolvency. 

50.-4  dividend. 
51.  ^n  assessment. 


II.  Notation. 


III.  Important  Facts. 


IV.  The  Five 
General  Cases. 


1.  /*er  cm^  ivritten  as  a  decimal. 

2.  Per  ce?i^  written  as  a  fraction. 

3.  77ie  commercial  sign. 

1.  0/  wAa^  factors  the  percentage  is  a  pi'oduct. 

2.  Of  what  the  per  cent  is  a  quotient. 

3.  Of  what  the  base  is  a  quotient. 

4.  Of  what  factors  the  amount  is  a  product. 

5.  Of  what  factors  the  difference  i^  a  product, 

6.  How  to  determine  a  partner^ s  share 
of  a  dividend  or  an  assessment. 

7.  Corresponding  terms  in  multiplication 
and  percentage. 

'  1.  To  find  the  percentage. 

2.  To  find  the  per  cent.  C  a.  Percentage  and  per  cent  givea 

3.  To  find  the  base.       ■<  b.  Amount  and  per  cent  given. 

4.  7h  find  the  amount.  L  c.  Difference  and  per  cent  given. 
^5.  7h  find  the  difference. 


276 


SECOND    BOOK  IN  ARITHMETIC. 


V.  Special  Applications 

OF   THE 

Five  Generax  Cases. 


1.  Profit  and  loss. 

2.  Commission.  .  •  -j  r 

3.  Insurance.  ^ 

J  T^  .-'    (  «.  Ad  valorem. 

5.  Duties,  i  7    c      .n 

„    „.    1     \^'  Specific. 

6.  Stocks.  ^        ^ 

7.  Partnership. 


On  money  collected. 
On  money  expended. 


B,  With  Time  as  an  Element. 


I.  Terms. 


'■);: 


Interest. 
Principal. 


3.  Amount. 

4.  Hate  of  interest. 


n.  Important  Facts. 


1' 

(3. 


1.  Upon  what  tlie  per  cent,  in  interest,  dep&nde. 
Corresponding  terms  in  interest  and  percentage. 
Of  what  three  factors  interest  is  a  product. 


III.  Computations. 


'5.  \   b. 

U.  : 


Interest  for  years. 
1.  Cases.  ■{  b.  Interest  for  months. 
Interest  for  days. 

ia.  Interest  for  1  year. 
b.  Interest  for  2  or  more  years. 
c.  Interest  for  any  other  time. 
d.  For  amount. 


SUPPLEMENT. 


§  1.   METHODS   OF  PROOF. 

1,  The  fundamental  rules  of  arithinetic  are  addition, 
subtraction,  multiplication,  and  division. 

All  arithmetical  computations  are  performed  by  the  use  of  one 
or  more  of  these  processes. 

The  methods  of  proof  of  the  fundamental  rules,  here  giv- 
en, are  based  upon  the  following  truths : 

I.  The  sum  is  not  changed  hy  changing  the  order  ia  wJuch 
the  parts  are  added. 

II.  A  whole  equals  the  sum  of  all  its  parts. 

III.  The  difference  between  one  of  tico  numbers  and  their 
su'in  equals'  the  other  number. 

IV.  The  product  is  not  changed  by  changing  the  order  of 
the  factors. 

V.  The  product  of  two  factors  divided  by  either  factor 
equals  the  other  factor. 

VI.  The  divisor  and  quotient  are  the  factors  of  the  divi- 
dend. 

2*  To  Prove  Addition. 
First  Method. — Add  the  given  parts  hi  reverse  order. 
The  two  results  must  be  alike.    (See  I.) 

Second  Method. —  Omit  one  or  more  of  the  parts,  and  to  the 

sum  of  the  remaining  parts  add  the  sum  of  the  parts  omitted. 

This  result  and  the  result  first  obtained  must  be  alike.    (See  11^ 


278  S  UP  PL  EM  EN  T. 

3.  To  Prove  Subtraction. 
First  Method. — Add  the  subtrahend  and  remai7ider. 

The  result  must  equal  the  mmuend.  (See  II.) 
Second  Method. — Subtract  thd  remainder  from  the  minuend. 

The  result  must  equal  the  subtrahend.    (See  III.) 

4.  To  Prove  Multiplication. 
First  Method. — Midtiply  the  multiplier  by  the  multiplicand. 

The  two  results  must  be  alike.    (See  IV.) 
Second  Method. — Divide  the  product  by  either  factor. 
The  result  must  equal  the  other  factor.    (See  V.) 

S*  To  Prove  Division. 
First  Method. — Multiply  the  divisor  and  quotient  together. 
The  result  must  equal  the  dividend.     (See  VI.) 
In  integers  arid  decimals^  add  any  final  remainder  to  tJie  product. 

Second  Mctliod. — Divide  the  dividend  by  the  quotient. 

The  result  must  equal  the  divisor.    (See  V.) 
In  integers  and  decimals,  subtract  any  final  remainder  from  the 
dividend,  before  dividing. 

Problems. 
Add  and  prove 

1,  2,416;  13,892;  937;  65,429;  182,705;  and  89,076. 

2,  95.73;  4.067;  238.4;  25.206;  317.0317;  and  4,404.0404. 
S.  I,  I,  if,  ih  H,  and  i. 

Subtract  )  4.  $2,719.37  from  $21,050.      6.  83.83  less  8.0383. 
and  prove  )  5,  505,992  less  278,059.  7.  fgi  -  f^. 

Multiply  )     S.  4,971  by  628.  10,  23.4  times  $189.18f. 

and  prove)     P.  32.07x5.213.  11. 

Divide  and  prove 


4¥T  -^  TrriU* 


12.  4,692  by  69. 

13.  5,831  by  84. 


U.  24.952-^4.76. 
15.  $293.75-r-45|. 


16.  $518.70by  $14.25. 

17.  m  by  if. 


[279] 

§  2.    GENERAL  PHINCIPLES  OF  DIVISION. 

6.  Any  change  in  either  dividend  or  divisor  produces  a 
change  in  the  quotient. 

1,  With  a  given  divisor,  the  greater  the  dividend  the  greater  the 
quotient ;  and  the  less  the  dividend  the  less  the  quotient. 

2.  With  a  given  dividend,  the  greater  the  divisor  the  less  the 
quotient ;  and  the  less  the  divisor  the  greater  the  quotient, 

7.  The  quotient  of  72  divided  by  12  is  6.     Then 

11-  - 

A.  12)72x2  12)72X3  12)72-^2  12)72 -r- 3 

12  18  3  2         ^^^^' 

Multiplying  the  dividend  multiplies  the  quotient;  and 
Dividing  the  dividend  divides  the  quotient, 

1  £  i  i 

B.  12X2)72  12X3)72  12-7-2)72  12-^3)72 

3  2  12  18         ^^^^' 

Multiplying  the  divisor  divides  the  qtwtient ;  and 
Dividing  the  divisor  multiplies  the  quotient. 

L  1  -  - 

C.  2X12)2X72    3X12)3X72    iofl2)iof72    j  of  12)^  of  72 

6  6  6  6       ^^^^' 

Multiplying  dividend  and  divisor  by  the  same  number  does  not 

change  the  quotient;  and 

Dividing  dividend  and  divisor  by  the  same  number  does  not 

change  the  quotient. 

S*  General  Principles  of  Division. 
I,  Multiplying  the  dividend  or  dividing  the  divisor  m/iil- 
tiplies  the  quotient. 

XL  Dividing  the  dividend  or  multiplying  the  divisor  di- 
vides the  quotient. 

III.  Multiplying  or  dividing  both  dividend  and  divisor  by 
the  same  number  does  not  change  the  quotient. 


[280] 

§  3.   GENERAL  PRINCIPLES  OF  FRACTIONS. 

9»  Any  change  in  either  term  of  a  fraction  produces  a 
change  in  the  value  of  the  fraction. 

1.  With  a  given  denominator,  the  greater  the  numerator  the 
greater  the  value  of  the  fraction;  and  the  less  the  numerator  the 
less  the  value  of  the  fraction. 

2.  With  a  given  numerator,  the  greater  the  denominator  the 
less  the  value  of  the  fraction ;  and  the  less  the  denominator  the 
greater  the  value  of  the  fraction. 

10.  The  value  of  the  fraction  ^  is  f     Then 

2  2  3  4 

.  Jg  X  g  _  g  15X3__S  1S^S__  1  13^S_  1      TT 

-^*  73  «  7S  /?  75  12  7S      —  1R       -"61106, 


'18 

Multiplying  the  numerator  multiplies  the  fraction;  and 
Dividing  the  numerator  divides  the  fraction. 

J^  2  3  4 

n         jg  _  J        _JL-—1  13   _i  13  __i    TT 

■'^'  73X3^13  73XS~'18  73-^3'"  3  73-^3       3      ^^^^^y 

Multiplying  the  denominator  divides  the  fraction;  and 

Dividing  the  denominator  multiplies  the  fraction. 

j^  2  3  4 

^         13X2  _  1  13X3  _  1  13^3  _  1  13^3  _  1      tt 

^*        73X3"'  6  73X3'"  6  73^3^  6  73^3~~  6      •"■^"^^' 

Multiplying  both  terms  by  the  same  number  does  not  change  the 
value  of  the  fraction  ;  and 

Dividing  both  terms  by  the  same  number  does  not  change  the 
value  of  the  fraction. 

11.  General  Principles  of  Fractions. 

I.  Multiplying  the  numerator  or  dividing  the  denominatof 
midtiplies  the  fraction. 

II.  Dividing  the  numerator  or  multiplying  the  denomina* 
tor  divides  the  fraction. 

III.  Midtiplying  or  dividing  both  terms  by  the  same  num* 
her  does  not  change  the  value  of  the  fraction. 


[281] 


§  4.   SHORT  METHODS  OF  COMPUTATION. 

Case  I.     The  multiplier  a  composite  number. 

12.  Ex.  Multiply  83  by  35. 
Explanation. — Since  35  is  5  times  7,  35 
times  any  number  is  5  times  7  times  that 
number.  I  therefore  multiply  83  by  7 
and  the  result  by  5,  ajid  obtain  2,905,  the 
required  result.     Hence, 

Rule. 
Multiply  successively  by  any  set  of  factors  of  the  number. 

Problems. 
1,  Multiply  4,285  by  the  factors  of  72. 


Process. 
35=:5x  7 
8S  X  7  =  581 
581  x5  =  2,905 


2.  42,196  by  108. 

8,   805,061  by. 252. 

i,  5.  878  by  21,  and  by  99. 

Case  II. 


6.  30.54  by  42. 

7.  500.05  by  6.Q. 

8.  .00964  by  .032. 


Tlie  multiplier  an  aliquot  part. 
13»  An  aliquot  part  of  a  number  is  any  exact  divisor 
of  tbe  number. 

5,  3^,  2i,  and  2  are  aliquot  parts  of  10. 

14.  Tbe  tmit  of  an  aliquot  part  is  the  number  divided 
to  obtain  the  part. 

100  is  tbe  unit  of  the  aliquot  parts  50,  33^,  25, 16f,  12^,  etc. 

IS.  Table  of  Aliquot  Parts. 


ift.. 

1  doz.     1  lb. 

UXITS. 

1 

10 

100 

1,000 

2,000  lb. 

or 
12  in. 

or         or 
12      16  oz. 

1  a. 

One  half 

^ 

5 

50 

500 

1,000  lb. 

6  in. 

6     8oz. 

80  sq. 

rd. 

One  third 

J 

3i 

33i 

333i 

6661'' 

4  " 

4 

One  fourth 

^ 

^ 

25 

250 

500   '' 

3  '' 

3 

4  " 

40      ' 

One  fifth 

^ 

2 

20 

200 

400   '' 

32      ' 

One  sixth 

6^ 

If 

16| 
12i 

1661 

333i-" 

2  " 

2 

One  eighth 

i 

H 

125 

250   *' 

2  '' 

20      * 

One  tenth 

To 

1 

10" 

100 

200    " 

16      * 

One  twelfth 

tV 

8| 

83^ 

1    '' 

1 

etc. 

282  SUPPLEMENT, 


i     3i  is  i  of      10,     3i )  r  i  of      10  )    times 

Since]    25    is  i  of     100,    25    p^°^^f  ^"7  )  i  of     100  (     that 
(  500    is  i  of  1,000,  500   S  ^^^^^^  ^^  (  i  of  1,000  )  number. 


r  8  qt.  arc  \  bu..  ^  the  r  8  qt.  ^  (  4  ^^  )  ^he  (  1  bu. 
I  6  in.  are  I  ft.,  V  cost -?  6  in.  Ws  ^  i  of  >  cost  •<  1  ft. 
(  400  lb.  arc  J  T.,    )   of    (  400  lb.  )      (  ^  of  )   of    ( 1  T 


Since 

Hence, 

16.  Rule. 
Multiply  by  the  unit  of  the  aliquot  part,  and  divide  the 
product  by  the  number  of  aliquot  parts  in  the  unit. 

Problems. 
What  is  the  product  of 

h  3J  times  582?        I     3.  491x500?    I     5.  2.015x2,500? 

^.  25  times  IjYSS?     |     4.  364xU?      I     ^.7.14x165? 
How  much  must  I  pay 

7.  For  12|  bu.  of  chestnuts,  at  $3.25  a  bushel  ?  (13^=  J  of  lOO) 

8,  For  83 J  A.  of  land,  at  $75  an  acre  ?  (^J  =  iV  of  1,000) 
P.  For  198  bar.  of  apples,  at  $3.33j  a  barrel? 

10,  For  1  grt.  gro.  of  clothes-pins,  at  Ij  cents  a  dozen? 

11,  For  625  bu.  of  potatoes,  at  $.75  per  bushel? 

12,  For  376  bu.  of  corn,  at  $1.12^  a  bushel? 

Case  III.     The  multiplier  9,  99,  or  any  number  ex- 
pressed by  9's. 

10—1=9;  100—1=99;  and  so  on.  Consequently,  10  times  any  num- 
ber, less  the  number,  is  9  times  the  number;  100  times  any  num- 
ber, less  the  number,  is  99  times  the  number  ;  and  so  on.    Hence, 

m.  Rule. 
Annex  as  many  O^s  to  the  multiplicand  as  there  are  9^s  in 
the  multiplier,,  and  subtract  the  midtip)licand  from  this  result. 

Problems. 

Jf.  80,207  by  99,999. 
5.  227,641  by  999,999. 
(  3.  16,971  by  9,999.  6.  26,942,781  by  99,999. 


i  1.  7,496  by  99. 
Multiply  \  2.  6,842  by  999. 


SHORT  METHODS   OF  COMPUTATION.      283 

Case  IV.    The  divisor  a  composite  number. 

IS.  Ex.     Divide  2,905  by  35.  ^ 

J.  ROCESS 

Explanation.— Since  35  is  5  times  7,  ^^g  <?  /r  _  r      7 

of  any  number  is  \  of  -|  of  that  num-  >  ^ 

ber.    I  therefore  divide  2, 905  by  5  and       2,905-^5  —  581 
the  result  by  7,  and  obtain  83,  the  re-  581-^1  —  88 

quired  result.     Hence, 

Rule. 
Divide  successively  hy  any  set  of  factors  of  the  number. 
Problems. 
i.  40,600  by  the  factors  of  56. 


Divide  H-\'^^^\^^- 

S,  179,412  by  48. 

.  4,  27,030  by  5.4. 


5,  1,720.32  by  168. 

6,  100.625  by  1.25. 

7,  87,444  by  2,520. 


Case  V.     The  divisor  an  aliquot  part. 

C  3^  is  ^  of  10,  )  any  [  3J  3  times  \  as  many  c  10. 
Since  •<    25    is  ^  of    100,  [•  number  •<    25   4  times  [•  times  as   \     100. 

(  500   is  J  of  1,000,  )  contains  (  500    2  times  )  it  contains  ( 1,000. 

C 1  bu.  is  4  times  8  qt. ,  \  the  ( 1  bu.  is  4  times  \  the  (  8  qt. 
Since  \  1  doz.  is  3  times    4,        [•  cost  \  1  doz.  is  3  times  [•  cost  \     4. 

(IT.  is  5  times    4001b.,)   of   ( 1  T.  is  5  times     )   of   (4001b, 

19.  Rule. 
Divide  hy  the  unit  of  the  aliquot  part,  and  multiply  the 
quotient  hy  the  numher  of  aliquot  parts  in  the  unit. 

Peoblems. 
What  is  the  quotient  of 


1,  4,280-r-2i-?      I    S,  66,000-^333j? 

2,  13,450-^500?  I    U.  119-f-16f? 


5.  $478.50-^121? 

6.  $35.48-r-$.20? 


Tj,.    -.      r  7.  Of  1  box  of  oranges,  at  $56.75  for  25  boxes. 
,    ^^       \8.  Of  1  yard  of  muslin,  at  $1.40j  for  12^  yards, 
tne  cost  I  <^    Qf  1  ^^^  Qf  g^g^l^  at  $8.12i  for  250  pounds. 

i(9.  $4.25        \       is  the       ^  yards  of  cloth,      \        ($1.25? 

11,  $13.18i   \    value  of    \  pounds  of  butter,  >•  at  -j  $.20? 

12.  $9,641      )  how  many  (  barrels  of  beef,     )        (  $16.66§ 


[284] 

§  5.   CONVERSE  REDUCTIONS. 

20*  Converse  operations  are  those  arithmetical  proc- 
esses that  are  the  reverse  of  each  other. 

a*  Subtraction  is  the  reverse  of  addition  ;  hence, 

Addition  and  subtraction  are  converse  operations. 

b.  Division  is  the  reverse  of  multiplication  ;  hence, 

Multiplication  and  division  are  converse  operations. 

Case  I.    A  decimal  to  a  fraction. 

21»  Any  decimal  may  be  written  in  two  forms. 
7  tenths  is  .7  or  ^^  ;  59  thousands  is  .059,  or  ^SSir ;   3  ten-thou- 
sandths is  .0003,  or  yjjj^jy. 

In  the  decimal  form,  the  unit  is  indicated  by  the  position 
of  the  decimal  point.  In  the  fractional  form,  it  is  expressed 
by  the  denominator,  which  is  1  with  as  many  ciphers  an- 
nexed as  there  are  decimal  places  in  the  decimal. 

Ex.  1.   Express  .125  in  fractional  form.  Process. 

Explanation. — I  write  the  number  without       225  =z  — —  z=  -^ 
the  decimal  point  for  a  numerator,  and      *  ^^^^      ^^ 

under  it  I  write  its  denominator,  1,000. 
Ex.  2.   Reduce  .083^  to  the  Process. 

fractional  form.  '083i-^f^,=f^  =  l 

Rule. 

Write  the  given  number  without  the  decimal  poi7it  for  a 
numerator,  and  under  it  write  its  denominator. 


Pboblems. 


U,  .1625=. what  fraction? 

5.  .00064  =  what  fraction  ? 

6.  .06875= what  fraction? 


1.  Reduce  .375  to  a  fraction. 

2.  Reduce  .9125  to  a  fraction. 
S.  Reduce  .85  to  a  fraction. 

7.  Reduce  18.75  to  a  mixed  fractional  number. 

8.  .00016  of  a  mile  =  w^hat  fractional  part  of  a  mile  ? 

9.  .02f  of  a  week  is  Avhat  fractional  part  of  a  week  ? 
10,  Reduce  $.16|  to  the  fraction  of  a  dollar. 


CONVERSE    REDUCTIONS.  285 

Case  II.    A  fraction  to  a  decimal. 

22»  Annexing  decimal  ciphers  to  the   numerator   of   a 
fraction  does  not  change  its  value. 

Ex.  Reduce  \\  to  the  decimal  form.  Peocess. 

Explanation.— I  first  reduce  the  nu-      n  __  ii-oooo  _  Qg'^g 

merator  11  to  ten-thousandths,  by  an-      ^^  ^^ 

nexing  four  decimal  ciphers. 
I  then  reduce  J-^ff^,  the  fraction  thus  formed,  to  a  decimal,  by 

dividing  its  numerator  11.0000  by  its  denominator  16. 
The  result,  .6875,  is  the  required  decimal. 

Rule. 

Aiinex  a  decwial  cipher  or  ciphers  to  the  iimnerator^  and 
divide  the  number  thus  formed  by  the  denominator. 

When  the  decimal  does  not  terminate  : — 
Write  a  fraction  after  the  decimal  figures  ;  or 

Carry  the  quotient  to  any  desired  number  of  decimal  places,  and 
annex  the  sign  +  to  it,  to  show  that  the  division  is  incomplete. 
Thus,J  =  .3i;  TT  =  -45A;  §  =  -666+;  f  =  .42857+. 


Problems. 


1.  Reduce  -^V  ^^  ^  decimal. 


2.  Reduce  4-|  to  a  decimal. 


3.  What  decimal  equals  -fi  • 
Jf-.  Change  y^'U'o  ^^  ^  decimal. 

5.  Reduce  18|  and  ^■^-^J  to  mixed  decimal  numbers. 

6.  4  of  a  day  is  what  decimal  of  a  day  ? 

7.  Reduce  19y\  yards  to  a  mixed  decimal  number. 

Case  III.     A  denominate   decimal  to   a   compound 
number. 

23,  Ex.  Reduce  .75  of  a  yard  to  a  compound  number. 
Explanation.— To  reduce  .75  of  a  Process. 

yard  to  feet,  I  multiply  by  3,  and        75  yd  X3  — 2  25  ft 

obtain  2.25  feet.  o  -  j-        i  q       q  nn  - 

To  reduce  the  .25  of  a  foot  to  inch-     -^^ /t'X2'<!  =  d.Ua  m. 

es,  I  multiply  by  12,  and  obtain  3  Heuce,  .75  yd.  =2  ft.  3  in. 

inches. 
Writing  the  integral  parts  of  the  results  in  order,  I  have  3  ft.  3  in,, 

the  required  compound  number. 


28G  SUPPLEMENT. 

In  this  process^  the  decimal  part  only  of  any  result  is  reduced 
to  a  lower  denomination.  In  other  respects  the  process  is  the  same 
as  the  general  process  of  Reduction  Descending  (Page  194). 

Problems. 
Reduce  to  compound  numbers 


i.  .8  da. 
2,  .625  yd. 


3.  .21875  mi. 
Jf.  21.875  bu. 


5.  .661- gal. 

6.  .0019f  sq.  yd. 


7.  .75  of  the  year  1882.       |         8.  |  of  a  gross  ton. 

Case  IV.  A  compound  number  to  a  denominate 
decimal. 

24.  Ex.  Reduce  2  ft.  3  in.  to  the  decimal  of  a  yard. 

Explanation. — To  reduce  3  inches  to  the       Process. 
decimal  of  a  foot,  I  annex  decimal  ciphers  12  \  3  00  in 

and  divide  by  12;  and  I  obtain  .25  of  a  — '- 1 

foot,  which  I  write  at  the  right  of  the  2  ^  I  2,25  ft, 

feet  of  the  given  number.  ,75  yd. 

To  reduce  the  2.25  feet  to  the  decimal    Hence,  2  ft,  3in,z=z,75  yd, 
of  a  yard,  I  divide  by  3,  and  obtain 
.75  of  a  yard,  the  required  decimal. 

In  this  procesSj  each  quotient  forms  the  decimal  pKi'^'t  of  the 
next  higher  denomination.  In  other  respects,  the  process  is  the 
same  as  the  general  process  of  Reduction  Ascending, 

Problems. 
Reduce  to  a  denominate  decimal  of  the  next  higher  de- 
nomination 

1,  3  pk.  5  qt.  1  pt.  of  grass  seed. 

2,  3  qt.  1  i  pt.  of  wine. 

3,  7  doz.  and  6  buttons. 
Jf,  4  yd.  1  ft.  8  in. 


5.  13  quires  6  sheets. 

6.  203  lb.  8  oz. 

7.  15  h.  50  min.  24  sec. 

8.  2  yd.  2  ft.  3  in. 


Case  V.     A   denominate    fraction    to    a    compound 
number. 

25.  Ex.  f  of  a  day  equals  what  compound  number  ? 

Explanation. — To  reduce  |  of  a  day  to  hours,  I  multiply  by  24, 
and  obtain  14|  hours. 


CONVERSE  REDUCTIONS.  287 

To  reduce  the  f  of  an  hour  to  minutes,  I  multiply  by  60,  and  ob- 
tain 24  minutes. 

Writing  the  hours  and  min-  Process. 

utes    of    these    results,   in  ^  da.x^i^^^  h.—  IJft  h, 

order,  I  have  14  hours  U  2j^y^eO  —  ^U-  min.  =2Jf  min. 

minutes,  the  required  com-  -i  i  -l   o  i 

pound  number  «^^^^'  i  ^^'  =  ^^  ^'  ^^  ^^^- 

The  processes  in  Cases  V  and  VI  are  essentially  the  same  as  the 
general  process  of  Reduction  Descending, 

Problems. 
What  compound  number  is  equivalent  to 


1.  y^  of  a  gross  ? 

2.  f  of  a  square  mile  ? 

3.  ^  of  an  acre  ? 


Jt-.  Iff  bushels? 

5.  i^yf  of  a  hogshead  ? 

6.  -|  of  a  leap-year  ? 


Case  VI.     A   coinpountl   number   to   a  denominate 
fraction. 

26*    Ex.  Reduce  14  h.  24  min.  to  the  fraction  of  a  day. 

Explanation. —  To  reduce  24  Process. 

minutes  to  the  fraction  of  an    p  /  ,„  •       .  f^n^z^  h  —^  h 
hour,  I  divide  by  60;  and  I    ^f  ^.^^^^  J^  "^  "  '.  %  f  "  f/l 
obtain  f  of  an  hour,  which  I    ^  4  /i.  +  j  h.  =  lJf^ti.=z  J^  ti, 
annex  to  the  14  hours,  mak-  -V  A.  -^-  ^4  =  f  da. 

ing  14|  hours.  Hence,  IJf,  h.2Jf.  min.  =  §  da. 

To  reduce  the  14|  hours  (=  ^f  h.) 
to  the  fraction  of  a  day,  I  divide  by  24,  and  obtain  |  of  a  day,  the 
denominate  fraction  required. 

The  minutes  in  14  hours  24  minutes  may  be  made  the  numerator, 
and  the  minutes  in  1  day  the  denominator  of  a  fraction.     Thus, 

14  h.  24  min.-=z864  min.; 

1  da.  =  24  h.  =  1,440  min.;  and 

864  min.=  ^\  da.=  f  da. 

Problems. 
Reduce  to  denominate  fractions  the  compound  numbers 


1.  2  qt.  1  pt.  3  gi. 

2.  24  sq.ft.  18  sq.  in. 

3.  130  rd.  2  yd.  1  ft.  3  in. 


4..  3  pk.  1  pt. 

5.  24  cu.  ft.  1,080  cu.  in. 

6.  8  h.  10  min.  21  sec. 


[288] 


§  6.  PRICE,  QUANTITY,  AND  COST. 

27*  In  all  transactions  of  purchase  and  sale,  and  of  labor 
and  wages,  four  elements  are  considered  —  viz.,  price,  the 
unit  of  price,  quantity,  and  cost, 

28*  Price  is  the  sum  paid  or  allowed  for  a  unit,  or  for  a 
fixed  number  of  units  of  the  commodity. 

20*  The  unit  of  price  is  the  number  of  units  upon 
which  the  price  is  based. 

The  unit  of  price  may  be  1  dozen,  1  hundred,  1  thousand,  1  ton,  or 
1  of  any  kind  or  denomination. 

30*  Quantity  is  the  number  of  units  or  parts  of  a  unit 
of  the  commodity. 

31.  Cost  is  the  sum  paid  or  allowed  for  the  quantity. 

Case  L    Price  and  quantity  given,  to  find  cost. 

32.  Examples.     Find  the  cost 

1.  Of  8i  days'  work,  at  $2.25  per  day. 

2.  Of  380  tomato  plants,  at  $1.75  per  hundred. 

3.  Of  15,968  feet  of  lumber,  at  $16.50  per  thousand. 

4.  Of  8,385  pounds  of  iron  castings,  at  $40  per  ton. 

Processes. 
Ex.  1,     8^x$2.25— $19,12^, 
P      g    {  S80  plants-^  100=3.80  hundred  plants  ;  and 

\  3.8 X  $1.75  =$6.65. 
-p,      ^  j  15,968  ft.-^  1,000=15.968  thousand  ft.  ;  and 
•        (  15.968  x$16.50=$263472. 

j  8,385  lh.-^2,000 =1^.1925  T. ;  and 
^''*  '^*  ( 11925 X  $40=  $167.70. 
Explanations,  r  in  Ex.  1  is         1  ^  ^^^  r  is    8i 

The  unit  J  m  Ex.2  is      100,  I  ^^-^tu„_  J  is     3.8. 
of  price   1  in  Ex.3  is  1,000,  \l^^^Z\  ''  ^^'^^^' 
I  in  Ex.  4  is  2,000,  J  ^*  ''''^^^  I  is    4.1925. 


PRICE,  QUANTITY,  AND    COST.  289 

I  therefore  find  the  cost,  in  each  example,  by  multiplying  the  price 
by  the  quantity,  ^.  e.  by  the  number  of  units  of  price.     Hence, 

Rule. 
Reduce  the  quantity  to  units  of  price,  and  multiply  the 
price  by  this  result. 

Problems. 
Find  the  cost 

1.  Of  429  barrels  of  flour,  @  $7.06^. 

2.  Of  91.88  A.  of  land,  @  $112.50. 

3.  Of  I  yd.  of  satin,  @  $1.75. 

4.  Of  3,145  fence  pickets,  @  $2.25  per  C. 

5.  Of  1,155  lb.  of  beef,  @  $14.50  per  C. 

6.  Of  15,690  ft.  of  lumber,  @  $18.75  per  M. 

7.  Of  85,432  bricks,  @  $7.50  per  M. 

8.  Of  2,784  pounds  of  hay,  @  $13  per  T. 

9.  Of  4,680  lb.  of  fertilizer,  @  $27.50  per  T. 

Case  II.     Price  and  cost  given,  to  find  quantity. 

S3,  Examples.     Find  the  quantity  that  can  be  bought 

1.  Of  tea  for  $30.73,  at  %.bQ  per  pound. 

2.  Of  beef  for  $44.27^,  at  $11.50  per  cwt. 

3.  Of  bricks  for  $110.25,  at  $8,75  per  thousand. 

4.  Of  merchant  iron  for  $125.46,  at  $85  per  ton. 

Processes. 
^       y   {  At  $1  a  poimd,  $30.73  will  buy  30,73  pounds  ;  and 
*  (  30. 73 pou nds -^.56=z 5Jf^ pounds. 
(  $11.50-^100— $.115,  the  price  of  1  pound  ; 
Ex.  2.  \  At  $1  a  pound,  $4-4-^7^  will  bmj  JiJf..27^  pounds ;  and 
(  44.27^  pounds -^.115 =385 pounds. 
(  $8.75-~-lfi00=:$.00875,  the  price  of  1  brick; 
Ex.  3.  \At$l  a  brick,  $110.25  will  buy  110.25  bricks;  and 
(  110.25  bricks-^  .008.75=12,600  bricks. 
$85-^2,000=z$.0425,  the  price  of  1  pound ; 
Ex.  ^.  "I  ^^  $i  a  pound,  $125.Jf6  luill  buy  125.46  pounds  ;  and 
125.46  pounds -^.0425 =2,952  2)ounds. 

N 


290 


SUPPLEMENT. 


Explanations. 

iin  Ex.  1   is         1,  ^        ,  r|.56. 

in  Ex.  3  is     100,  I  ^^^/^^  J  ^\^  of  $11.50.  or  $.11^ 
in  Ex.  3  is  1,000,   (     P"^?    1  txtW  of  l^-^.To,  or  $.00875. 
in  Ex.  4  is  1   T.,  J   ^^  -^  ^^   IWiRr  of  $85,  or  $.0425. 
I  therefore  find  the  quantity,  i.  e.  the  number  of  units  of  price,  by 
dividing  the  cost  by  the  price  of  1.     Hence, 

KULE. 

Find  the  price  of  1,  by  dimding  the  price  of  a  unit  by  the 
unit  of  price  ;  and  then  divide  the  cost  by  the  price  of  1, 

Problems. 

"  barrels  of  flour,  at  $7.06j  per  barrel  ? 

tons  of  iron,  at  $56.25  per  ton  ? 

yards  of  ribbon,  at  $|  per  yard  ? 

fence  pickets,  at  $2.25  per  hundred? 

pounds  of  beef,  at  $11.50  per  cwt.  ? 

bricks,  at  $7.50  per  thousand  ? 

feet  of  lumber,  at  $18.75  per  thousand  ? 

pounds  of  hay,  at  $13  per  ton  ? 
^  pounds  of  fertilizer,  at  $27.50  per  ton? 

Case  III.     Quantity  and  cost  given,  to  find  price, 

54.  Examples.     Find  the  price 

1.  Of  1  sewing-machine,  at  $2,372.50  for  Qb  machines. 

2.  Of  transporting  1  cwt.  of  express  freight  from  Chicago 
to  New  York,  at  $56.10  for  2,125  lb. 

3.  Of  1  thousand  bricks,  at  $148.67^^  for  15,650  bricks. 

4.  Of  a  ton  of  hay,  at  $19.14  for  2,640  pounds. 

Processes. 

$2,372.50-^-65  =.$36.50. 

{2,125  lh.-r-100=i21.25  cwt;  and 

\  $56.10-^21.25  =z$2.6i. 

j  15,650  hricJcs-^  1,000 z=z  15. 650  thousand  bricks  ;  and 
(  $U8.67i-r- 15.65  =$9.50. 


i.  $3,029.8li^ 

2.  $10,336.50 

is  the 

S.  $2|| 

4.  $68.78^ 

cost 

5.  $44,275 

► 

6.  $640.74 

of  how 

7.  $294.18|- 

8.  $18,096 

many 

9.  $64.35 

Ex.  1. 

Ex.  2. 

Ex. 


PRICE,    QUANTITY,  AND    COST 


291 


•  (  $1, 


^^-  •*•  i$19.U. 
Explanations. 


and 


-1.32=:$U.50, 


Ex. 
Ex. 


The  quantity  |  . 
or  number    j  .^  ^^   3 
of  units       [i^Ex.  4 


, ,,         ,     r  is  $3,373.50 ; 
and  the  cost    J  53  igg.io  ; 
-  of  the  number  <  . 


of  units 


is  65, 
is  21.25, 
is  15.650, 

is     1.320,  J         ----^-         Lis  $19.14; 
I  therefore  find  the  price,  in  each  example,  by  dividing  the  cost  by 
the  quantity,  ^.  e.  by  the  number  of  units  of  price.    Hence, 

Rule. 
Beduce  the  quantity  to  units  of  price,  and  divide  the  price 
by  this  result. 

Pkoblems. 

$3,029|f  for  429  bar.  of  flour, 
$10,336.50  for  183.76  T.  of  iron, 


i. 

2. 
S. 

4. 
5. 
6. 

7. 
8. 
9. 


3. 81 J  for  -|  bar.  of  kerosene, 


how 


much 


for 


barrel  ? 

ton? 

barrel  ? 

hundred  ? 

cwt.  ? 
1  thousand? 
1  thousand? 
1  ton? 
1  ton? 


3d*  Formulas. 


$68.78^  for  3,057  fence  pickets, 
$44,275  for  385  lb.  of  beef, 
$640.74  for  85,432  bricks, 
^94y\  for  15,690  ft.  of  lumber, 
18.096  for  2,784  lb.  of  hay, 
34.35  for  4,680  lb.  of  fertilizer. 

Case  I.     Price  X  quantity  z=z  cost. 
Case  XL      Cost  ~  price       =  quantity. 
Case  III.     Cost -r- quantity  z:^  price. 

Problems. 

1.  3  lb.  8  oz.  of  opium  costs  how  much,  @  |4.75  per  lb.  ? 

2.  At  $4.75  per  lb.,  how  much  opium  costs  $16.62^? 
8.  If  3^  lb.  of  opium  costs  $16|,  what  is  the  price? 

Jf.  755  broom  handles  cost  how  much,  @  $1.12|-  per  C? 

5.  At  $4.50  per  bar.,  how  much  kerosene  costs  $2.81  J? 

6.  13,450  bricks  cost  how  much,  at  $6.50  per  M  ? 

7.  $5.62|  buys  how  much  wood,  at  $7.50  per  cd.  ? 

8.  How  much  must  I  pay  for  3,575  lb.  of  coal,  at  $6.50 
per  T.  ? 


292  SUPPLEMENT. 

9,  $61.68f  for  27^^  gross  of  buttons,  is  how  much  per 
gross  ? 

10.  ^237.82  for  12,826  barrel  staves,  is  how  much  per  C? 

11.  How  much  must  a  potter  pay  for  5,040  pounds  of 
clay,  at  $16.50  per  T.  ? 

12.  $250  pays  for  how  much  gas,  @  $2.50  per  M  ft.  ? 

13.  $60.50  for  968  lb.  of  grapes,  is  how  much  per  T.  ? 
H.  Find  the  cost  of  59|^  bu.  of  wheat,  @  $.93}  per  bu. 

15.  $258  buys  how  much  himber,  <d  $24  per  M  ? 

16.  How  many  tons  of  fertilizer  can  be  bought  for 
$111.56*^,  at  $28.33^  per  ton? 

17.  A  fruit  dealer  sold  47  bu.  1  pk.  2  qt.  of  chestnuts,  at 
$3.25  a  bushel.     How  much  did  he  receive  for  them? 

18.  The  water  in  a  mill  flume  1}  rd.  long,  10  ft.  wide,  and 
6  ft.  deep  weighs  54  T.  281:^  lb.  How  much  does  1  cu.  ft. 
of  water  weigh  ? 

19.  I  paid  $8.10  for  35  lb.  of  ice  three  times  per  week 
from  April  16  to  Oct.  1.     What  was  the  price  per  C  ? 

20.  A  hardware  merchant  sold  1,952  pounds  of  steel  for 
$256.20.     What  was  the  price  per  ton  ? 

21.  A  furniture  maker  paid  $258.75  for  ash  lumber,  at 
$45  per  M.     How  much  did  he  buy  ? 

22.  Find  the  cost  of  12  pieces  of  French  calico,  averaging 
36  yd.  each,  at  $.16f  a  yard. 

23.  A  tanner  bought  a  hide  that  weighed  97|  pounds,  at 
9Jc.  a  pound.     How  much  did  he  pay  for  it  ? 

2Jt..  Five  loads  of  coal  with  the  wagon  weighed  3,219  lb., 
3,074  lb.,  3,621  lb.,  3,342  lb.,  and  2,990  lb.;  and  the  wagon 
weighed  964  lb.  What  was  the  value  of  the  coal^at  $5.75 
per  ton  ? 

25.  If  one  ton  of  ore  yields  1,276.5  pounds  of  copper,  how 
much  ore  will  yield  857.8  pounds  of  copper? 


[293] 


§  7.  COMPOUND  NUMBERS. 

I.  Money  Values. 

80*  3Ioney  is  the  legal  or  recognized  standard  of  the 
measure  of  value. 

Money  consists  of  coins,  made  of  gold,  silver,  or  other  metal. 

37 •  A.  coin  is  a  piece  of  metal  on  which  certain  charac- 
ters are  stamped,  by  authority  of  the  General  Government, 
making  it  legally  current  as  money. 

United  States  coins  are  made  of  gold,  silver,  nickel,  and  bronze. 

88*  Alloy  is  a  baser  metal  mixed  with  a  finer. 

a.  Pure  gold  or  silver  coins  would  be  so  soft  as  to  perceptibly 
depreciate  by  wear.  Hence,  gold  and  silver  coins  are  alloyed, 
to  increase  their  hardness. 

b.  Gold  is  alloyed  with  silver,  and  silver  with  copper. 

c.  The  United  States  gold  and  silver  coins  are  .9  by  weight  pure 
metal,  and  .1  alloy.  The  alloy  of  gold  coins  consists  of  9  parts, 
by  weight,  of  silver  and  1  part  of  copper  ;  and  that  of  silver 
coins, .  1  of  pure  copper. 

d.  The  bronze  cent  has  .95  of  copper  and  .05  of  tin  or  zinc  ;  the 
3-cent  and  5-cent  pieces,  .75  of  copper  and  .25  of  nickel. 

39*  Paper  fttoney  is  a  legal  or  recognized  substitute  for 
coin. 

a.  Treasury  notes  are  notes  or  bills  issued  by  the  General 
Government. 

b.  JBanlc-notes  or  hank-bills  are  notes  or  bills  issued  by  a 
banking  company.     (See  111,  page  316.) 

40.  Currency  is  the  coin  and  paper  money  in  circulation 
in  trade  and  commerce. 

Coin  is  commonly  called  specie  currency.  Treasury  notes 
and  bank-notes  are  commonly  called  papei^  currency. 


294 


SUPPLEMENT. 


41. 

Government  Table. 

10  mills  are  1  cent. 

10  cents  *'    1  dime. 

10  dimes  *'    1  dollar. 

10  dollars  <'    1  eagle. 


Uniteil  States  Money* 

Oold  coins.— DonhlQ  eagle,  eagle,  half  eagle, 

3-dollar  piece,  quarter  eagle,  dollar. 
Silver  coins,— DoWav,  half  dollar,  quarter  dol- 
lar, dime. 
Nickel  coin. — 5-cent  piece. 
Bronze  coin. — Cent. 
Money  value  is  commonly  expressed  in  dollars  and  cents,— eagles 
being  expressed  as  dollars,  ditnes  a^  cents,  and  mills  as  fractions  or 
decimals  of  a  cent. 

43.  French  Money  is  the 

money  of  France. 

The  denominations  are  the 
franc  (fr.),  the  decime  (dc),  the 
centime  (ct.),  and  the  millime 
(m.). 


42,  English  Money  is  tlie 
money  of  Great  Britain. 

The  denominations  are  the 
pound  (£),  the  shilling  (s.), 
tha  penny  (d.),  and  the  farthing 
(far.  or  qr.). 

4  far.  are  1  d. 
12  d.      "    1  8. 
20  8.       "    1  £. 
A  sovereign  is  an  English 
gold  coin  whose  value  is  £1. 

44.  Canada  Money  is  the 

money  of  the  Dominion   of 
Canada. 

The  denominations  are  the 
dollar  and  the  cent. 

100  cents  are  1  dollar. 
20  cents  are  a  shilling,  and  5 
shillinf^s  a  dollar. 


10  m.  are  1  ct. 
10  ct.  •'  1  dc. 
10  dc.   "    1  fr. 

The  value  of  a  franc  is  $  .193. 

43.  German    Money    is 

the    money   of    the    German 
Empire. 

The   denominations    are   the 
pfennig  (penny),  and  the  mark. 
100  pfennige  are  1  mark. 
The  value  of  a  mark  (Reichs- 
marken)  is  23.85  cents. 


II.  Measures. 

4G.  Surveyors^  measures  are  the  measures  used  by  sur- 
veyors in  measuring  the  boundaries,  and  in  estimating  the 
areas  or  square  contents  of  lands. 

The  denominations  are — {1.)  For  linear  measures,  the  mile^  the 
chain  (ch.),  and  the  link  (1.) ;  and  (2.)  For  square  measures,  the 
township,  the  square  mile  or  section,  the  acre,  the  square  chain, 
square  rod  or  pole  (P.),  and  the  square  link. 


COMPOUND   NUMBERS. 


295 


4:7*  A  Gunter's  chain 

sists  of  100  links,  each  7.92 
by  surveyors  in  measuring 

Linear  Measures. 

100  1.      are  1  cli. 

80  ch.     "1  mi. 

Square  Measures. 
625  sq.  1.     are  1  P.  or  sq.  rd. 

16  P.  '*    1  sq.  ch. 

10  sq.  ch.    "    1  A. 
640  A.  *'    1  sq.  mi.,  or  sec. 

36  sq.mi.   "    1  Tp. 


is  4  rods  or  66  feet  long,  and  con- 
inches  long.  This  chain  is  used 
the  boundaries  of  land. 

a.  Since  100  links  are  1  chain,  1  link  is 
.01  of  a  chain,  35  links  are  .35  of  a 
chain ;  and  so  on.  Hence,  links  may 
be  written  either  as  hundredths  of  a 
chain,  —  thus,  15.44  ch. ;  or  chains 
and  links  as  a  compound  number  ; 
thus,  15  ch.  44  1. 

b.  Since  a  chain  is  4  rods  long,  25  links 
are  1  rod.  The  denomination  rod  is 
seldom  used  in  linear  chain  measure. 


6 

5  1  4 

3 

2 

1 

7 

8   9 

10  1  11 

12 

18 

17 

16 

15  14 

13 

19 

20 

21 

22  23 

24 

30 

29 

28 

27  26 

25 

31  32 

33 

34  j  35 

36 

c.  Civil  engineers  on  railroads  and  canals  use  an  engineers^ 
chain,  which  consists  of  100  links  each  1  foot  long. 

48,  Govermnefit    lands    arc 

commonly  divided  by  parallels  (east 
and  west  lines),  and  meridians  (north 
and  south  lines),  into  townships  6 
miles  square. 

In  the  same  way  each  township  is 
divided  into  36  sections  or  square 
miles,  each  section  into  4  quarter- 
sections  of  160  acres  each,  and  each 
quarter-section  into  4  :5-quarter-sec- 
tions  of  40  acres  each. 

a.  For  easy  designation,  convenient 
reference,  and  ready  location  on 
maps  of  government  surveys,  town- 
ships are  numbered,  in  order,  from 
a  given  parallel  or  base  line,  north ; 
and  ranges  of  townships  from  a  giv- 
en meridian,  east  or  west. 

b.  Tp.  25  N.  of  R  6  W.  is  the  25th 
township  north  of  a  principal  base 
line,  and  in  the  6th  range  of  town- 
ships west  of  a  principal  meridian.  A  Section,     l  mile  square. 


A  Township.     6  miles  square. 


N.  }^  Section. 
320  Acres 


S.  W.  X 

160  Acres 


of 


N.E.3^ 
of 

S.E.^ 


S.E.i^ 

of 
S.E.^ 


296  S  UPFL  EM  EN  T. 

49*  Mariners  use  the  following  denominations  : 
6  ft.  are  1  fathom,  in  measuring  depths  at  sea. 

120  fathoms    "    1  cable's  length,  for  short  distances. 
1.15  mi.        "1  knot,  or  nautical  mile. 
3.45  mi.       "    1  league,  or  3  nautical  miles. 

SO.  In  f/eographical  and  astronomical  calculations 

1  geographic  mi.  is  1.15  statute  mi. 

3         "  "  are  1  league.    (1.) 

60  "  "  or  )    *'   1  degree  (deg.)  of  latitude,  or 

69.16  statute     '*       )  of  longitude  on  the  equator. 

a*  The  knot  is  used  in  measuring  the  speed  of  vessels. 

b.  The  nautical  mile  (or  knot)  and  league  are  the  same  as  the 
geographic  mile  and  league. 

c.  The  length  of  a  degree  of  latitude  is  not  quite  uniform.  69.16 
miles  is  the  average  length,  and  is  the  one  adopted  by  the  U.  S. 
Coast  Survey. 

d.  In  measuring  the  height  of  horses,  4  inches  are  1  hand,  the 
measure  being  taken  directly  over  the  shoulder. 

e.  In  measuring  the  length  j   6  points  are  1  line, 
of  clock  pendulums,  (  12  lines  are  1  inch. 

f.  In  measurins:  the  i  «  v  i  ,  •  i. 
1  i^i  r  XT  ^  X  i  3  barleycorns  or  sizes,  are  1  inch, 
length  of  the  foot,  (             ' 

g'.  The  sacred  cubit  mentioned  in  the  Bible  is  21.888  inches. 

h.  The  old  road  measures,  40  rd.  are  1  furlong  (fur.),  and  8  fur. 

are  1  mi. ,  are  now  but  little  used. 

Circular  and  Angular  Measures. 

Sl»  A  fnatJiematical  circle  is  the  area  bounded  by  a 
circumference.     (See  254.) 

S2»  A  geographical  circle  is  the  circumference  of  a 
mathematical  circle. 

S3.  An  angle  of  1  degree  is  1  of  the  360  equal  angles 
that  exactly  fill  the  space  about  a  common  point  in  a  plane. 

S4:>  A  geographical  degree  is  1  of  the  360  equal  parts 
of  a  geographical  circle. 

Angles  are  at  a  centre,  and  degrees  are  parts  of  a  circumference. 
Hence,  angles  are  not  degrees. 


COMPOUND    NU3fBBRS.  297 

dS*  The  measure  of  an  angle  at  the  centre  of  a  circle 
is  that  part  of  the  circumference  included  between  the  sides 
of  the  angle. 

Since  circles  may  be  great  or  small,  the  degrees  in  their  circum- 
ferences will  be  correspondingly  great  or  small.  An  angle  of 
1  degree  is  constant;  while  the  measure  of  the  angle — i.e.,  1  de- 
gree in  a  circumference — varies  with  every  change  in  the  cir- 
cumference of  a  circle. 

SO*  Circular  and  angular  measures  are  used 

1.  By  surveyors,  in  determining  the  directions  or  bearings 
of  land  boundaries  and  other  lines. 

2.  By  navigators,  in  determining  the  position  and  course  of 
vessels  at  sea. 

S.  By  geogj^aphers  and  astronomers,  in  determining  lati- 
tude and  longitude ;  in  determining  the  position  and  motion  of 
the  heavenly  bodies ;  and  in  computing  difference  in  time. 

The  denominations  of  circular  and  angular  measures  are 
{1.)  For  surveyors  and  astronomers, — the  circumference,  the  quad- 
rant (quad.),  the  degree  (°),  the  minute  ('),  and  the  second  C) ; 
and 
{2.)  For  astronomers, — the  circle,  the  sign  (S.),  the  degree,  the  min- 
ute, and  the  second. 


Surveyors  and  Navigators' 

Measures. 
60"  are  V. 

60'  "    1°. 

90°  ''1  quad. 

4  quad.      **    1  circumference. 


Astronomers'  Measures. 
60"     are  V. 
60'       ♦<    1°. 
30°       *'    1  S. 
12  S.    "    1  Circle. 


III.  Weights. 

87*  Avoirdupois  tveights.     Denominations  in  common 
nse : 

of ^  ^.^'  r.  .   ^    T  i  ^'  °^  '??*  !■  at  the  New  York  State  Salt  Works. 
S80  lb.  (5  bu.)     ♦'    1  bar.       <'    ) 

100  lb.  "1  cental  of  grain. 

100  lb.  ♦<    1  cask  of  nails  or  raisins. 

N  2 


298 


SUPPLEMENT. 


58.  Table  of  Avoirdupois  Pounds  in  a  Bushel, 
As  fixed  by  statutes  in  the  States  named. 


Barley 

Beans 

Buckwheat.  . 
Clover  seed  .  . 
ludian  corn  .  , 

Oats 

Potatoes. .  .  .  , 

Rye , 

Timothy  seed 
Wheat 


50 
40  45 


O  Q 


52  56  56 

28 

60 

56 


32 
54 
60  56  60 


4848  48 
6060  60 
4050  52 
6060  60 
5256  56 
323235 
60|60|60 
5456  56 
45|4o45 
60'60'60 


48 
60 
52 
60 
56 

100  to 
3  bu. 

60 
56 
45 
60 


46 

46 

.56 
30  32 

56  56 

60!60 


^  L*  ,>^  1.2 


I 


;z;  [Jzj  l:z;  jo  io  ^^  >  |^'  ^ 
48'40  47  46  45  48 


_l_ 


48  48  48 

60 

50  50  481 

64|60; 
54  56  58; 
30  30  32 

60;60 

5656 
44 

60  60 


|42^48 


60,60 


46  42  42 
160  60 


60 


56|56,56^56'56!56 
32  34  32J 


^1^ 


32  36.32 

|60|     |60|6060 

56  56  56  56!56.56 


X 


,  ,  ,  ,  1^ 

60  60  60  60  60  60 


S9*  Troy  tveigJUs  are  the  weights  used  in  weighing 
the  precious  metals  and  jewels,  and  in  philosophical  experi- 
ments. 


The  denominations  are  the  pound,  the 
ouTice,  the  pennyweight  (pwt.),  and  the 
grain  (gr.). 


24  gr.    are  1  pwt. 
20  pwt.    *♦    1  oz. 
12  cz.       '*    1  lb. 


GO*  Apothecaries^  iveighfs  and  measures  are  weights 
and  measures  used  by  physicians  in  prescribing,  and  apoth- 
ecaries in  mixing,  medicines. 

The  denominations  are  the  pound,  the  ounce  (oz.  or  I ),  the  dram 
(dr.  or  3 ),  the  scruple  (sc.  or  3),  and  the  grain. 

Apothecaries'  Weights. 

20  gr.    are  1  sc.  or  3 

3  9       ♦'    1  dr.  or  3 

8  3        ''1  oz.  or  I 

12  z        Ml  ii,_ 


Apothecaries'  Fliiid  Measures. 
CO  minims  (or  drops)  are  1  fluid  drachm. 

8  fluid  drachms  "    1  fluid  ounce. 

16  fluid  ounces  "    1  pint. 

8  pints  <<    1  gallon. 


61»  The  Government  standard  units  of  the  money, 
measures,  and  weights  now  in  use,  and  from  which  the 
other  denominations  in  the  respective  tables  are  deter- 
mined, are  given  on  the  following  page. 


LONGITUDE  AND    TIME. 


299 


TABLES. 

United  States  Money, 
Lines,  Surfaces,  and  Solids, 
Liquid  Measure, 
Dry  Measure, 
Troy  Weight, 
Avoirdupois  Weight, 


Dollar, 
Yard, 
Gallon, 
Bushel, 
Pound, 
Pound, 
For  ordinary  purposes,  2150.4  cubic  inches  are  called  a  bushel. 

IV.    COMPAEATIVE    YaLUES. 


VALUBS. 

.800  silver,  .100  alloy. 
3  feet,  or  36  inches. 
231  cubic  inches. 
2,150.42  cubic  inches. 
5,760  grains. 
7,000  Troy  grains. 


I.  Of  measures 
of  capacity. 


DENOMINATIONS. 

1  gal., 
Iqt, 
1  pt, 

DENOMINATIONS. 


n.  Of  weights.  \\^^'' 
\  1  oz.. 


LIQUID  MEASURE. 

231  cu.  in., 

57.75       "     " 
28.875     "     " 

TROY  WEIGHT. 

5,760  gr., 
480    " 


DRY   MEASURE. 

268.8  cu.  in.  (.5  pk.) 
67.2    "    " 
33.6    "    " 

AVOIRDUPOIS  WEIGHT. 

7,000    gr. 
437.5  '' 


a.  Multiplying  the  number  of  cubic  inches  231  268.8 
in  a  liquid  gallon  by  7,  and  the  number               *^  g 

in  a  dry  gallon  by  6,  we  find  that  7  liquid      — '- — -        

gallons  contain  4.2  cubic  inches  more       Ijol?  1,612.8 

than  6  dry  gallons.     Hence,  in  ordinary  computations,  it  is  suf- 
ficiently accurate  to  estimate  7  liquid  gal.  =  6  dry  gal. 

b.  Multiplying  the  number  of  grains  in  a  pound  Troy  by  175, 
and  the  number  in  a  pound  avoirdupois  by  144,  we  have 

5,760  X 175  =  7,000  x  144.     Hence, 
175  pounds  Troy  =144  pounds  avoirdupois. 


§  8.  LONGITUDE  AND  TIME. 
03 •  The  earth  revolves  east  on  its  axis  once  in  24  hours, 
and  the  sun  appears  to  pass  west  round  the  earth  in  the 
same  time.  That  is,  the  sun  appears  to  pass  west  over  15° 
in  1  hour,  over  15'  in  1  minute,  and  over  15"  in  1  second. 
Hence, 

The  relative  po8ition  to  the  sun,  of  any  place  on  the  earth,  deter- 
mines the  time  at  that  place. 

A  difference  of  15°  in  longitude  makes  a  difference  of  1  h.  in  time. 
4.  15'  («  a  a  1  min.     ♦* 

44  15"  •*  "  "  Isec.       " 


300 


SUPPLEMENT. 


64.  Longitude  is  reckoned  east  or  west  from  a  given 
meridian.  The  meridian  selected  for  this  purpose  is  called 
the  prime  tneridian.  The  meridian  generally  used  is 
that  which  passes  through  the  Astronomical  Observatory, 
Greenwich,  England.  Some  nations  also  reckon  longitude 
from  the  capital  of  their  own  country. 

63 •  Longitude^  from  Greenicichy  of  H  important  cities. 


Cities. 

Longitudes. 

Cities. 

Longitudes. 

Washington, 

77°    0'  15"  W. 

Berlin, 

13°  23'  45"  E. 

Philadelphia, 

75°  10'          W.    ! 

Canton, 

113°  14'           E. 

New  York, 

74°    3'          W. 

Cincinnati, 

84°  39'  21"  W. 

Boston, 

71°     3'30"W.    1 

Chicago, 

87"  37'  45"  W. 

London, 

5'  38"  W.    I 

New  Orleans, 

90°    2'  30"  W. 

Paris, 

2°  20'          E. 

St.  Louis, 

90°  15'  15"  W. 

Rome  (Italy), 

12°  27'          E. 

San  Francisco, 

122°  26'  45"  W. 

Case  I.  Difference  in  time  given,  to  find  difference 
in  longitude. 

00.  Ex.  The  difference  in  time  between  Washington  and 
London  is  5  h.  V  min.  46  sec.  What  is  the  difference  in 
longitude  ? 

Process. 


h.  7  min.  1^.6  sec. 
15 


Explanation. — Every  second  of  difference 
in  time  makes  15"  of  difference  in  longi- 
tude ;  every  minute  of  difference  in  time, 
15'  of  difference  in  longitude ;  and  every 
hour  of  difference  in  time,  15°  of  differ-  76'°  55-'  gQ'f 
ence  in  longitude.     Since  either  factor 

may  be  used  as  the  multiplier,  I  multiply  5  h.  7  min.  4G  sec.  by 
15,  and  obtain  76°  56'  30",  the  required  difference  in  longitude. 

Rule. 
Multiply  the  difference  in  time  by  15. 

The  product  will  be  the  difference  in  longitude. 

a.  Mid-day,  or  any  other  given  time,  occurs  sooner  at  any  place 
east,  and  later  at  any  place  west  of  a  given  point.     Hence, 

b.  Time-pieces  are  faster  at  any  place  east,  and  slower  at  any 
place  west,  of  a  given  point,  than  they  are  at  that  point. 


LONGITUDE  AND    TIME.  301 

Pkoblems. 
Find  the  difference  in  longitude  between  two  places,  the 
difference  in  time  being 

1,  32  min.     \     3.     6  min.  20  sec.    I     5.  1  h.    8  min.  30  sec. 

2.  54  sec.      I     ^.28  min.  45  sec.     |     ^.  8  h.  12  mm.  27  sec. 

Find  the  longitude  of  a  place  whose  time,  compared  with 


Greenwich  time,  is 

7.  13  min.  20  sec.  slow. 

8.  1  b.  10  rain.  fast. 

9.  6  h.  40  min.  12  sec.  slow. 


Washington  time,  is 

10.  2  h.  fast. 

11.  24  min.  52  sec.  slow. 

12.  9  h.  9  min.  9  sec.  fast. 


Case  II.  Difference  in  longitude  given,  to  find  dif- 
ference in  time. 

67.  Ex.  The  difference  in  longitude  between  Washing- 
ton and  London  is  76°  56'  30".  What  is  the  difference  in 
time  ? 

Explanation.— Every  15"^  of  differ-  Process. 

ence  in  longitude  makes  1  li.   of    .r\>yfionf^p  onff 

difference   in  time;    every  15'   of    J-^)  J^    '^^>  ^^ 

difference   in  longitude,  1   min.  of  5  h.  ?  min.  Jf.  u  nee. 

difference  in  time  ;   and  every  15" 

of  difference  in  longitude,  1  sec.  of  difference  in  time.  I  there- 
fore divide  TB""  56'  30"  by  15,  and  obtain  5  h.  7  min.  46  sec,  the 
required  difference  in  longitude. 

Rule. 
iJlvskle  the  difference  in  longitude  by  15. 
The  quotient  Avill  be  the  difference  in  time. 

Pkoblems. 
Find  the  difference  in  time  between 


1.  Washington  and  Paris. 

2.  Philadelphia  and  Rome. 

3.  London  and  New  Orleans. 


Ji..  St.  Louis  and  Canton. 

5.  Cincinnati  and  Berlin. 

6.  Greenwich  and  Washino^ton. 


7-10.  Noon  at  )  (New  York,)  find  the  (Boston. 

11-H.  1  h.  30  min.  a.m.  at)  (Chicago,     S  time  at  (San  Francisco. 


[302] 

§  9.  METRIC  SYSTEM  OF  MEASURES  AND  WEIGHTS. 

68.  A  meter  is  a  measure  of  length. 

69*  The  metric  system  employs  the  meter  as  the  uni- 
form standard  unit  of  all  measures, — whether  of  length, 
area,  volume,  weight,  or  money  value. 

70.  A  metric  number  is  a  number  by  which  units  of  suc- 
cessive denominations  of  measure,  weight,  or  money  value, 
are  expressed  in  the  decimal  scale,  or  scale  of  10. 

Standard  Units. 

71*  The  tneter,  the  liter,  and  the  gram  are  the  three 
standard  units  of  the  metric  system. 

a.  The  meter  is  the  nnit  of  measures  of  length. 
It  is  39.37+  inches  long  (or  about  39i  inches). 

b.  The  liter  is  the  unit  of  measures  of  volume. 

It  is  a  cube  whose  edge  is  one  tenth  of  a  meter,  or  3.937  inches. 

c.  The  grafn  is  the  unit  of  weight. 

It  is  the  weight  of  a  cube  whose  edge  is  one  hundredth  of  a  meter, 
or  .3937  of  an  inch. 

72,  Surfaces  and  volumes  are  simply  the  squares  and 
cubes  of  the  measures  of  length. 

The  ar  is  another  name  for  the  square  decameter,  or  100  square 
meters  of  land ;  the  ster  is  another  name  for  the  cubic  meter  of 
fire-wood ;  and  the  tonneau  is  another  name  for  the  weight  of  a 
cubic  meter  of  w^ater.     Hence, 

73*  Every  possible  dimension  (length,  area,  volume),  ca?^ 
he  measured  with  the  meter;  every  p)ossible  capacity  icith 
the  liter ;  and  every  possible  iveight  with  tlie  gram. 

Multiples  and  Divisors  of  the  Standard  Units. 

74.  The  names  of  the  multiples  of  the  standard  units 
are  formed,  by  placing  before  the  names  of  the  standard 
units  the  Greek  prefixes  deha,  liehto,  Mlo. 


METRIC  MEASURES  AND    WEIGHTS. 


303 


The  names  of  the  divisors  of  the  standard  units  are  formed 
by  placing  before  the  names  of  the  standard  units  the  Latin 
prefixes  deci,  centif  fnilli, 

a.  Deka  signifies  10;  hekto,  100;  and  kilo,  1,000. 

b.  Deci  signifies  .1 ;  centi,  .01 ;  and  milli,  .001. 

7^0    Spelling,   Pronunciation,   and    Abbreviations    of 
THE  Metric  Units. 


Multiple  Prefixes. 

Spell-  Pronun-  Abbre- 

ing.  ciation.  viation. 

Deka  deka  D. 

Hekto        hekto  H. 

Kilo  kilo  K. 

Myria        miria  M. 


Standard  Units. 

Pronun- 
ciation. 

meeter 
leeter 
gram 
ar 


Spell- 
ing. 

meter 

liter 

gram 

ar 

ster 

tonneaH 


Abbre- 
viation. 

m. 
1. 


St. 


Divisor  Prefixes. 

Promin-    Abbre- 
ciation.    viation. 

dese         d. 

sente       c. 

mill         m. 


Spell- 
ing. 

deei 
centi 
milli 


In  writing  metric  numbers, 
the  names  of  the  places  used 
correspond  in  meaning  to  the 
English  names  of  the  same 
places  used  in  writing  mixed 
decimal  numbers.     Thus, 


o 

'^. 

-^^ 

g.S 

M 

o 

o  ^ 

0 

0 

0 

0. 

CO 

02 

02 

;-i 

-^ 

^ 

fl 

O 

f^ 

O 

o 

^ 

-«-9 

w  ^ 

Tli 

Si-B 

i 

§§ 

a  a 

Extension. —  7    6 


Volume, —     7    6 


76.  All  the  multi- 
ples and  divisors  of 
metric  numbers  are  in 
the  scale  of  10;  they 
may  be  written  as  here 
showm. 

These  numbers  may  be 
read,  giving  name  to 
each  unit ;  thus,  7  Km. 
6  Hm.  5  Dm.  4  m.  3  dm. 
2  cm.  1  mm. ;  etc. 
In  the  same  manner  read 

1.  The  numbers  above,  expressing  volume. 

2.  The  numbers  above,  expressing  weight. 


Weight. - 


bo     bo 

7    6 


0 

CQ 


^3 
O 


a 

a 

1  meters 


4.    3    2    1  liters 


4.    3    2 


bO 

a 

1  grams 


6    5 

4 

3 

2 

J  mm. 

7    6 

5 

4 

3 

2cm. 

.1 

7 

6 

5 

4 

3dm. 

.21 

7 

6 

5 

4m. 

.321 

7 

6 

5Dm. 

.4321 

7 

6Hm. 

.54321 

304  'S:  UP  PL  EM  EN  T. 

Any  one  of  these  numbers  may  be  expressed  in  units  of  any  one 
of  its  denominations,  and  read  as  ones  and  decimals  of  that  de- 
nomination.    For  example, 

The  numbers  expressing  extension  may  be  read  in  the  denom- 
ination of  the  standard  unit  (the  meter),  thus:  7,654  and  321 
thousandths  meters. 

Again, — this  number  may  be 
expressed,  either  as  millime- 
ters, centimeters,  decimeters, 
meters,  dekameters,  hekto- 
meters,  or   kilometers,  and 

read  accordingly. 

°  ^  7^™-  .654321 

77.  In  measuring  and  tveighing, 

Measure  all  dimeiwions  in  meters^  all  capacities  i?i  liters, 
and  all  toeights  in  grams,  icsing  decimals  only, 
7S»  In  computations^  use 

1.  Ar  and  hectar  for  area  of  land ; 

2.  Square  meter,  square  kilometer,  square  decimeter,  and  square 
centimeter  for  other  areas ; 

3.  Ster  for  solidity  of  wood ; 

Jf,  Cubic  meter,  cubic  decimeter,  and  cubic  centimeter,  for  other 
volumes ; 

5.  Tonneau  for  the  weight  of  articles  now  estimated  by  the  ton. 

70.  Metric  Unit  Equivalents. 
A  meter  is  ^J-  of  a  yard,  or  a  little  more  than  39.37  inches. 
A  liter  is  about  .9  of  a  dry  quart,  or  1^^  liquid  quarts, 
A  gram  is  about  ^  of  an  avoirdupois  ounce. 
A  kilogram  or  kilo  is  about  2.2  pounds. 
An  ar  is  3.95  square  rods, 
A  ster  is  35.32  cubic  feet. 
A  tonneati  is  2,204.6  pounds. 

SO.  The  successive  units  of  any  number  expressing  metric 
measure  or  weight  are  in  the  scale  of  10.  Multiples  of  any 
metric,  unit  are  expressed  as  integers,  and  divisors  as  deci- 
mals ;  and  the  metric  number  is  written  and  read  as  other 
integers  and  decimals.    Hence,  metric  tables  are  unnecessary. 


METRIC  MEASURES  AND    WEIGHTS.         305 

SI*  Rule  for  Metric  Notatiois^. 
Write  the  metric  number  the  same  as  any  other  numher  in 
the  decimal  scale,  loith  the  addition  of  the  symbol  indicating 
the  unit  in  which  it  is  to  be  read, 

82.  Rule  for  Metric  Numeration. 
Head  the  number  as  a  number  in  the  decimal  scale,  giving 
to  it  the  denomination  indicated  by  its  symbol.     Or, 

Head  the  units  of  each  place  separately,  beginning  at  the 
left,  and  calling  thousands  kilo,  hundreds  hekto,  te7is  deka, 
tenths  deci,  hundredths  centi,  and  thousandths  milli. 
Exercises. 

Ji.   9548"^-. 263 

5.  95""\48263 

6,  954^'"-.8263 
IS.  52^^78149 
IJf.  527^^8149 
15.  527^^-8M49 

Write 

16.  8  kilometers  7  hektometers  3  dekameters  5  meters. 

17.  b  meters  9  centimeters  4  dekameters  2  millimeters. 

18.  Twenty-eight  thousand  four  hundred  sixteen  and  five 
ten-thousandths  centimeters. 

19.  Six  liters  six  deciliters  six  centiliters  six  milliliters. 
20.^  Three  kiloliters  four  liters  five  and  seven  tenths  de- 
ciliters. 

21.  Forty-one  and  five  thousand  one  hundred  sixty-five 
ten-thousandths  kiloliters. 

22.  Thirty-seven  and  twenty-nine  hundredths  hektograms. 

23.  3,869  and  218  thousandths  kilograms. 

21/,.  Nine  kilograms  twenty-two  grams  three  and  forty- 
six  hundredths  decigrams. 

S3*  Metric  numbers  are  added,  subtracted,  midtiplied,  and 
divided  in  the  same  nnanner  as  other  decimal  numbers. 
Hence,  special  rules  for  these  processes  are  unnecessary. 


Read 

1. 

gKm.  gHm.  ^Dm.  4™.  gd.n.  gem.  2mm. 

2. 

5KI.  gHl.  3DI.  51.  2^11.  401.  gml. 

3. 

^Kg.  gHg.   |Dg.  9ff.  4dg.  ^cg.  3mg. 

7. 

g54Dm.  gm. 

10.   139^\4528 

8. 

1394^528 

11.   139^^1\528 

9. 

13"\94528 

12.  5,278M4c 

) 

[306] 


§  10.    INTEREST. 

1.  Legal  Interest. 

84.  Legal  rate  of  interest  is  the  rate  allowed  by  law. 

Sti,  Usury  is  any  rate  of  interest  greater  than  the  legal 
rate. 

In  most  States,  a  party  receiving  usury  is  liable  to  a  penalty. 

86.  Legal  Rates  of  Interest. 


states. 

Rates  %.    1 

states. 

Rates  %. 

states. 

Rates  %. 

Alabama 

8             1 

Kentucky 

6 

N.C. 

6  to  8 

Arizona 

10  to  any 

Louisiana 

5  to  8 

Ohio 

6  to  8 

Arkansas 

6  to  10 

Maine 

6  to  any 

Oregon 

8  to  10 

California 

7  to  aiy 

Maryland 

6 

Penn. 

6 

Colorado 

10  to  any 

Mass. 

6  to  any 

R  I. 

6  to  any 

Conn. 

6  to  any 

Michigan 

7  to  10 

S.  C. 

7 

Dakota 

7  to  12 

Minnesota 

7  to  10 

Tennessee 

6 

Delaware 

G 

Mississippi 

6  to  10 

Texas 

8  to  12 

D.  C. 

6  to  10 

Missouri 

6  to  10 

Utah 

10  to  any 

Florida 

8  to  any 

Montana 

10  to  any 

Vermont 

6 

Georgia 

7  to  8 

Nebraska 

7  to  10 

Virginia 

6 

Idaho 

10  to  18 

Nevada 

10  to  any 

Wash.  T. 

10  to  any 

Illinois 

6  to  8 

N.  H. 

6 

V/.  Va. 

6 

Indiana 

6  to  8 

N.J. 

6 

Wisconsin 

7  to  10 

Iowa 

6  to  10 

New  Mex. 

6  to  any 

Wyoming 

12  to  any 

Kansas 

7  to  12 

New  York 

6 

a.  If  no  rate  of  interest  is  specified  in  written  obligations  involv- 
ing interest,  the  legal  rate  is  always  understood. 

b.  The  legal  rate  in  any  State  in  which  two  rates  are  given  in  this 
table,  is  the  less  of  the  two  rates,  unless  a  higher  rate  is  specified. 
Any  rate  not  exceeding  the  higher  rate  given  is  legal,  if  stipu- 
lated in  writing. 

II.  Six  Per  Cent  Method. 

87*  At  Vlfo  per  annum,  the  rate  per  month  is  1^  ; 
At  6^  per  annum  {^  of  12^),  the  rate  per  month  is  ^^. 


INTEREST. 


307 


Rule. 

I.  To  find  the  per  cent : — Divide  the  time  expressed  in 
months  hy  2, 

II.  To  find  the  interest :  —  Multiply  the  principal  hy  the 
per  cent. 

Or,  since  at  12^  per  annum  the  rate  per  month  is  1^, — 
Divide  the  principal  hy  ^,  and  midtiply  the  quotient  hy  .01  of 
the  time  expressed  in  months. 

Problems. 
At  6^  per  annum,  what  is  the  per  cent 


i,  .^.  For  6  mo. ?    For  9  mo.? 
3.  For  8  mo.  12  da.? 
Jf.  For  5  mo.  18  da.? 


5.  For  1  yr.  6  mo.  ? 

6.  For  3  y  r.  1  mo.  15  da.  ? 

7.  For  2  vr.  29  da.? 


At  %fo  per  annum,  what  is  the  interest 


8-11.  Of  $250 

12-15.  Of  $1,913.50 

16-19.  Of  $629.37 

20-23.  Of  $12,078 


for 


4  mo.  9  da.  ? 

5  yr.  7^  mo.  ? 
4  yr.  20  da.  ? 

12  yr.  11  mo.  11  da.? 

Short  Method  for  Days  or  for  Months  and  Days. 
r  for  12  mo.  (1  yr.)  is  .06 

At  6^  per  annum,  J    ''60  da.  (^  yr.  or  2  mo.)  '*  .01 


the  per  cent 


6   ''  (iV  of  60  da.) 
1    "  (^  of  6  da.) 


.001 
.000^  or  i 


i.  e. ,  the  per  cent  is  6X)Vo  of  the  number  of  days.     Hence, 
SS*  Rule. 
Divide  the  number  of  days  hy  GfiOO,  and  multiply  the 
principal  hy  this  result. 

Or,  Multiply  the  principal  hy  a  fraction  whose  numerator  is 
the  number  of  days,  and  whose  denominator  is  6,000. 
Problems. 


At  6^,  what  is  the  interest  of 

1.  $753  for  20  da.? 

2.  $18,720  for  75  da.? 

3.  $2,869.25  for  33  da.? 

4.  $158.3li  for  12  da.? 


At  6^,  what  is  the  interest  of 

5.  $10,000  for  3  da.? 

6.  $500  for  18  da.? 

7.  $21,315  for  93  da.? 

8.  $50,000  for  1  da.  ? 


308  SUPPLEMENT. 


Find  the  interest  of 
9.  $240  for  2  mo.  11  da. 

10.  $476  for  6  mo.  15  da. 

11.  $853.25  for  3  mo.  3  da. 

12.  $2,681.39  for  8  mo.  10  da. 


Find  the  interest  of 

13.  $350.75  for  90  da. 

IJf.  $12,097  for  3  mo.  9  da. 

15.  $972  for  8  mo.  12  da. 

16.  $2,345.50  for  4  mo.  21  da. 


III.  The  Six  General  Cases  of  Interest. 

89*  The  per  cent  may  be  regarded  as  the  product  of 
the  numbers  expressing  rate  of  interest  and  time  (see 
3^3,  h.)  ;  L  e., 

Per  cent  =  rate  of  interest  x  timej  in  years. 

00,  The  terms  used  in  interest  correspond  to  the  terms  used 
in  percentage,  as  sliown  below. 


Terms  used  in  interest. 

Terms  used  in  percentage. 

Principal 

is 

base; 

Rate  of  interest 

is 

per  cent; 

Interest 

is 

percentage  ; 

A mou nt  { principal + interest) 

is 

amount  {base -^percentage) ; 

Difference  ( principal — interest) 

is 

difference  {base  —percentagt) 

Considering  the  terms  used  in  interest  as  interchangeable  with 
those  used  in  percentage — as  given  above, — 

The  six  general  cases  of  interest  are  readily  deduced  from  the 
five  general  cases  of  percentage. 

Case  I.     Principal,  rate  of  interest,  and  time  given, 
to  lind  interest. 

01.  In  percentage,  base  xj^er  cent  — the  percentage.   Hence, 

Formula. — Interest  n^principal  xper  cent. 
This  case  has  been  fully  considered  on  pages  257-264 

Case  II.     Interest,  rate  of  interest,  and  time  given, 
to  lind  principal. 

02*  In  percentage,  the  percentage  -^per  cent  —  base.    Hence. 

Formula. — Principal z=iinterest-r-per  cent. 
Ex.  $136.95  is  the  interest  on  what  sum  of  money  for  2 
yr.  9  mo.,  at  6^  ? 


INTEREST.  ^  309 

Explanation.— $136.95,  Process. 

the  interest,  is  the  per-    ^n  os>  -ion  m 

centao-e-and    165  the    ^  V^'d  mo.=z33  mo,  =  .165,per  cent; 
per  cent  on  $1  for  2  yr.     $1^6,95^.165^  $830,  principal 
9  mo.,  is  the  per  cent.  I 

therefore  divide  $136.95,  the  interest  or  percentage,  by  .165,  the 
,per  cent,  and  obtain  $830,  the  base,  which  is  the  required  principal. 

Problems. 
Find  the  principal,  the  interest  of  which 

1.  Is  $50.75,  at  5^,  for  2  yr.  6  mo. 

2.  Is  $1,748.50,  at  8^,  for  2  yr.  9  mo.  6  da. 

3.  At  6^,  for  6  mo.  10  da.,  is  $300. 

^.  At  seven  per  cent  per  annum,  is  five  cents  a  day. 

5.  For  2  yr.  4  mo.  12  da.,  at  6^,  is  $313. 

6.  At  4^  for  3  yr.  6  mo.,  is  $5.11. 

Case  III.      Principal,    rate    of    interest,    and    time 
given,  to  find  amount. 

95.  In  percentage,  base  x  (i-f^:>er  cent)  =iamoimt.    Hence, 
Formula. — Amount  =z principal  x  {l-\-per  cent). 
This  case  has  been  presented  on  pages  242,  243. 

Case  IV.     Amount,  rate  of  interest,  and  time  given, 
to  find  principal. 

04z*  In  percentage,  amount -r-  {l-\-per  cent)  —  base.     Hence, 
YoRUUL A.  —  Principal— amount-^  [l-f-per  cent). 

Ex.  What  sura  of  money,  at  interest  2  yr.  9  mo.,  at  6^, 
will  amount  to  $966.95  ? 

Explanation. —  Process. 

$966.95,     the       ^        g  mo.  =33  mo.  =  .165, per  cent; 
amount    of    the  i   ,     i /- ^       t    -/ ^  r         *     Ji 

required    princi-  1  ■\- J65  =  1 .  IbO,  ami.  of  1 ; 

pal  for  2  yr.  9  $966 .95 -^  1 .  165  =  $830, principal. 
mo. ,  at  6^  per  an- 
num, is  the  amount ;  and  1.165,  the  amount  of  1  +  the  per  cent 
on  1  for  2  yr.  9  mo.,  is  the  amount  of  1  for  the  given  time  at  the 
given  rate.  I  therefore  divide  $966.95,  the  given  amount,  by 
1.165  or  by  1  plus  the  per  cent,  and  obtain  $830,  the  base,  which 
is  the  required  principal. 


310  .        SUPPLEMENT. 

Problems. 
What  principal  will  amount 
i.  To  $500.55  in  6  mo.,  at  7^? 

2,  To  $356  in  1  yr.  4  mo.,  at  6^? 

3,  To  1720.60  in  2  yr.,  at  8^? 
i.  To  $135  in  10  yr.,at  10^? 

5.  To  $1,660.10  in  1  yr.  25  da.,  at  4^? 

6,  To  $461.12  in  1  yr.  11  mo.  18  da.,  at  5^? 

Case  V.     Principal,  interest,  and  time  griven,  to  find 
rate  of  interest. 

Od*  In  percentage, 

The  percentage-^  {base  xper  cent  at  1^)  =per  cent.    Hence, 

Formula. — Bate  of  int€rest=interest-h{principal  xper  cent  at  1^). 

Ex.  At  what  rate  of  interest  will  $400  gain  $60,  in  2  yr. 
6  mo.? 

Explanation.—  Process. 

$400,  the  prin-  ^        ^  ^,  ^^- 

cipal,      multi-  ^  ^^-  ^  ^o.=2i  yr.  =  .025,per  cent  at  1^ ; 
plied  by  .025,  $Jf00x,025=$10; 

the  per  cent  on  qqx ^^^^-^A^e^, 

$1  for  2  yr.  6  ^  lo^     lO'         ^ 

mo.  at   1^,   is 

$10.     Since  $10  is  tlie  interest  at  1^  for  the  given  time,  $1  is 

the  interest  at  -^  oil'fo,  and  $60  must  be  the  interest  at  60  times 

^  of  1^,  or  fj^,  or  6^,  which  is  the  required  rate  of  interest. 

Problems. 
Find  the  rate  at  which  the  interest 

1.  Of  $36.50  for  4  yr.  5  mo.  26  da.  is  $7,373. 

2.  Of  $278  for  3  yr.  5  mo.  12  da.  is  $47.95f 
S,  Of  $311.50  for  1  yr.  4  mo.  is  $24.92. 

^.  Of  $57.92  for  3  yr.  7  mo.  9  da.  is  $12.54. 

5.  Of  $273.51  for  2  yr.  20  da.  is  $50.23. 

6.  Of  $1,950  for  5  yr.  4  mo.  is  : 


INTEREST.  311 

Case  VI.  Principal,  interest,  and  rate  of  interest 
given,  to  find  time. 

06*  Per  cent— rate  of  interest  X  time  ;  and 
Interests  principal  x  rate  of  interest  X  time.     Hence, 

Formula. — Time  — inter  est -r-  {principal  x  rate  of  interest). 

Ex.  In  what  time  will  $400  gain  $60,  at  6^? 

Explanation.— $400,  the  prin-  Process. 

cipal,  multiphed  by  .06,  the  tl.on^  n^-tP/  • 

rate  of  interest,  is  $24,  the  ^^  ^-^^^  ^  '^^  -  ^"^-^^ 
interest  on  the  principal  for  -^  yr.  =  ^|-  yr.  —  2  yr.  6  mo. 
1  year.  If  $1  were  the  inter- 
est for  1  year,  $60  would  be  the  interest  for  60  years.  But 
since  $24  is  the  interest  for  1  year,  $60  is  the  interest  for  ^  of 
60  years,  which  is  fj  years,  or  2 J  years,  or  2  yr.  6  mo.,  the  re- 
quired time. 

Problems. 
'  1.  Will  $125  gain  $13.29  interest,  at  6^? 

2.  Will  $150  gain  $18.75,  at  5^? 

3.  At  4  J^,  will  $100  gain  $100? 
i.  Will  any  sum  double  itself,  at  1^  a  month? 
5.  At  10^,  will  $15,000  lose  $750? 

.  6.  Will  $340.25  gain  $20.84,  at  7^? 

Problems  in  the  Six  General  Cases  of  Interest. 

1.  The  interest  on  a  note  at  9^,  for  1  yr.  8  mo.,  is  $42. 
For  what  sum  was  the  note  given  ? 

2.  May  5,  I  borrow  $1,725  in  Detroit,  Mich.,  and  pay  it 
Nov.  15,  with  interest.     How  much  do  I  pay  ? 

S.  A  note  for  $600.50  and  interest  is  dated,  Cleveland,  O., 
July  26, 1881.     What  amount  is  due  Jan.  26, 1884? 

i.  What  is  the  semi-annual  interest  of  a  mortgage  for 
$4,375,  on  a  house  in  Baltimore,  Md.? 

5.  Sept.  3,  1884,  a  Kansas  farmer  pays  $1,477.59,  the 
amount  of  a  mortgage  on  his  farm,  dated  April  7,  1882. 
For  what  sum  was  the  mortgage  given? 


In  what  time 


312  SUPPLEMENT. 

6.  What  sum  must  be  put  at  interest,  at  6^,  that  a  boy 
15  yr.  4  mo.  old  may  receive  $2,010.  when  21  years  old? 

7.  In  4  years  I  am  to  receive  $800.  What  sum  should  I 
receive  now  to  cancel  the  claim,  money  being  worth  5^  ? 

8.  A  man  in  Pittsburgh,  Pa.,  paid  $134.40  for  the  in- 
terest on  a  mortgage  of  $420.  How  long  had  the  mortgage 
run? 

9.  A  young  lady  had  $1,600  at  interest,  at  6^,  until  it 
amounted  to  $2,000.     What  was  the  time  ? 

10.  I  paid  $4.60  interest  on  a  loan  of  $276,  for  2  mo.  15 
da.     What  rate  of  interest  did  I  pay  ? 

11.  A  lady  whose  expenses  are  $1  a  day,  has  $9,125  in 
money.  At  what  rate  must  she  loan  it,  that  the  interest 
will  pay  her  expenses  ? 

12.  A  South  Carolina  planter  paid  $22.40  interest  on  a 
loan  for  9  mo.  18  da.     What  was  the  loan  ? 

13.  I  received  $98.28  interest  for  1  yr.  8  mo.  24  da.  on  a 
loan,  at  7^.     What  was  the  sum  loaned  ? 

14.  Oct.  31,  1882,  a  San  Francisco  lawyer  collected  a  debt 
of  $840,  which  had  been  due  since  May  29,  1880.  What 
was  the  amount  collected  ? 

15.  Philadelphia,  Jan.  28,  1882,  I  paid  $1,698.50,  the 
amount  of  a  note  dated  Sept.  18, 1878.  For  what  sum  was 
the  note  given  ? 

16.  April  4  of  this  year  a  Denver,  Col.,  merchant  paid  a 
note  dated  June  27  of  last  year,  for  $3,650  and  interest. 
What  amount  did  he  pay  ? 

17.  A  Missouri  lumberman  paid  $103.68  interest,  at  the 
highest  legal  rate,  for  a  loan  of  $684.     Whaf  was  the  time  ? 

18.  I  bought  a  pair  of  horses  for  $450,  and  sold  them  at 
20^  advance,  receiving  in  payment  a  note  due  in  1  yr.  3  mo. 
15  da.,  at  6^  interest.  Eight  months  afterward  I  sold  the 
note  for  $550.  What  rate  of  interest  did  I  receive  on  the 
note  ?  If  the  note  was  paid  at  maturity,  what  rate  of  inter^ 
est  did  the  buyer  of  the  note  receive  on  his  investment  ? 


INTEREST.  313 

IV.  Exact  Interest. 
07*  In  computing  interest  for  months  and  days  by  the 
methods  given  on  pages  257-264,  30  days  are  regarded  as  a 
month,  and  360  days  as  a  year.  Consequently,  the  interest 
for  months  and  days,  found  by  these  methods,  is  -^^  or  i^ 
part  of  itself  too  great,  for  a  common  year ;  and  -^^  or  -^ 
part  of  itself  too  great,  for  a  leap-year.     Hence, 

98.  Rule. 

I.  Compute  the  interest  for  months  and  days  by  the  general 
rule. 

II.  Prom  the  result  subtract  j^j  of  itself,  if  the  year  is  a 
common  year;  or  /^  of  itself  if  the  year  is  a  leap-year. 

^rk    T^  r,         '  S  Prin.x  rate  X  time  in  yr,  and 

Vlfm  roRMULA. — Axact  interest z:^  i  o/?r^7         o/^i?^z      ^ 

(  booths  or  oboths  of  a  year. 

Peoblems. 
What  is  the  exact  interest 

1.  Of  $84.25  for  10  mo.,  at  the  rate  in  Illinois? 

2.  Of  $400  for  2  mo.  9  da.,  at  the  rate  in  Indiana  ? 

5.  Of  $3,225  for  1  yr.  7  mo.  19  da.,  at  the  rate  in  Con- 
necticut ? 

4.  Of  $1,852.60  for  2  yr.  3  mo.  12  da.,  at  the  rate  in  Ken- 
tucky ? 

6.  Of  $7,325  for  3  yr.  2  mo.  20  da.,  at  the  rate  in  Virginia  ? 

6.  Of  $32,782.52  for  4  yr.  4  mo.  10  da.,  at  the  rate  in 
Massachusetts  ? 

7.  Of  $22,064.60  for  4  mo.  15  da.,  at  the  rate  in  New 
Jersey  ? 

5.  Of  $8,673  for  20  days,  at  the  rate  in  Pennsylvania? 
9.  Of  $15,000  for  1  day,  at  the  rate  in  ]S"ew  York  ? 

10.  Of  $2,000  from'  Nov.  15,  1880,  to  April  1, 1881,  at  5fo  ? 

11.  Of  $6,500  from  March  1, 1882,  to  Aug.  10  of  the  pres- 
ent year  ? 

12.  Of  $9,355.25  from  April  4  to  July  7,  at  1f^  ? 

O 


[314] 

§  11.  BUSINESS  PAPER. 

100.  Business  paper  consists  of  written  or  printed  doc- 
uments, used  as  representatives  of  money  value. 

The  principal  classes  of  business  paper  are  notes,  oi'ders, 
drafts,  and  receipts.     (For  forms,  see  pages  360-362.) 

101.  An  indorsement  of  any  business  paper  is  a  writing 
upon  it,  that 

1.  Assigns  or  transfers  ownership;  or 

2.  Gives  security  for  the  fulfilment  of  the  obligation;  or 

3.  Acknowledges  a  partial  payment. 

An  mdorsement  is  usually  written  on  the  back  of  the  paper. 

102.  Negotiable  paper  is  any  business  paper  that  may 
be  transferred,  with  or  without  indorsement. 

103.  A  2y^oniissory  note  is  a  paper  that  acknowledges 
a  debt,  and  promises  payment  at  a  specified  time,  uncondi- 
tionally. 

1.  The  maker  of  a  note  is  the  person  who  signs  it. 

2.  The  ijayee  is  the  person  to  whom  the  note  is  payable. 

3.  The  indorser  is  the  party  who  writes  his  name  upon  the 
back  of  the  note,  as  security  for  its  payment. 

4.  Days  of  grace  are  three  days  allowed  after  the  time 
named  in  the  note  has  expired,  before  the  note  is  legally  due. 

5.  Maturity  is  the  last  day  of  grace. 

a»  When  the  third  day  of  grace  falls  on  Sunday  or  on  a  legal  holi- 
day, the  note  matures  on  the  second  day  of  grace. 

b.  The  banks  of  Delaware,  Maryland,  Missouri,  Pennsylvania,  and 
Washington  City  charge  interest  for  the  day  on  which  a  note  is 
discounted,  and  the  day  on  which  it  matures, — making  the  in- 
terest period  practically  one  day  longer  than  in  other  States. 

6.  The  face  of  a  note  is  the  sum  to  be  paid  at  its  maturity. 

The  face  of  a  note  on  interest  is  the  amount  of  principal  and  in- 
terest. 


[315] 

§  IS.  DISCOUNT. 

104.  Discount  is  a  deduction  from  a  selling  price,  or 
from  an  account,  or  from  any  obligation  before  it  is  due. 

10 5.  The  2>re«eiif  ivortli  or  proceeds  of  an  obligation 
is  its  face  minus  the  discount. 

Discount   is   of  three   kinds,  commercial   discount,  bank 
discount,  and  true  discount, 

I.  Commercial  Discount. 
100*  Coniniercial  discount  is  a  percentage  deducted 
from  the  list  price  of  goods,  or  from  the  gross  amount  of 
bills  or  other  obligations,  without  regard  to  time. 

a.  Per  cent  offh  another  name  for  commercial  discount. 

b.  The  proceeds  or  net  value  of -a  bill  is  the  gross  amount 
minus  the  percentage. 

107*  Ex.  Bought  a  bill  of  goods  amounting  to  $487  at 
list  prices,  15^  off.     What  was  the  net  value  of  the  goods  ? 

Peocess. 

$48  7  X  .lo  zzz  $73.05,  commercial  discount;  and 

$Jf.87~$7S.05=z$4.13,95,  net  value ; 

Or, 

1  —  .15  —  .85,  net  rale;  and  $  ^8  7  y,. 8  5  ^$4  IS, 9  5, net  value. 

lOS*  In  computations  in  commercial  discount, 

a.  Invoice  price  or  amount  of  ohliffation=zbase  ; 

b.  Per  cent  off=zper  cent ; 

c.  Commercial  discount  =zpercentage. 

100 •      j  I.  Obligation  X^  off  =  commercial  discount. 
Formulas.  (  II.  Obligation — commercial  discount  =zproceeds. 
Or,  III.,  Obligation  X  net  rate ^=  proceeds  or  net  value. 

Problems. 
What  is  the  commercial  discount  on  a  bill  of  goods 

1.  Invoiced  at  $300,  sold  on  3  mo.,  2^^  off  for  cash  ? 

2.  Invoiced  at  $1,250,  sold  on  4  mo.,  5^  off  for  cash  ? 


316  SUPPLEMENT. 

3.  For  $862.50,  sold  on  90  da.,  Sj^  off  for  cash  ? 
Jf.  For  $219. V5,  sold  on  60  da.,  l^'jC  off  for  cash? 
Find  the  j  5.  A  book,  the  Hst  price  being  $2.50,  30;^  off. 
cost  of    (  ^.  A  box  of  glass,  hst  price  $8.90,  60^  and  25^  off. 

7.  1  gross  of  fruit  cans,  at  $1.50  per  doz.,  40;^  and  b</o  off. 

8.  A  case  of  boots  (12  pairs),  list  price  $33, 16§^  and  3;^  off. 
Find  the  commercial  discount,  and  the  cash  proceeds,  of 

invoices  of  goods  of  the  following  amounts : 

9.  Of  $852,  sold  on  60  days,  3^  off  for  cash. 

10.  Of  $972.83,  sold  on  3  months,  4j^  off  for  cash. 

11.  Of  $1,500,  sold  on  30  days,  commercial  discount  1|^. 

12.  Of  $2,450,  sold  on  90  days,  commercial  discount  3^. 

II.  Bank  Discount. 
110*  A  hank  is  a  chartered  institution  that  receives  and 
loans  money,  or  issues  bank-bills  that  circulate  as  money. 

a.  A  batik  of  issue  is  one  that  issues  notes  or  bank-bills. 

b.  A  hank  of  discount  is  one  that  lends  money,  by  dis- 
counting notes. 

c.  A  savings-hank  is  one  that  receives  money  on  deposit, 
paying  interest  on  the  sums  deposited;  and  loans  its  de- 
posits, by  discounting  notes  with  real-estate  securities. 

d.  Some  banks  combine  two  of  these  kinds  of  business ;  and 
some,  all  of  them. 

111.  Bank-hills  or  hank'notes  are  promissory  notes  is- 
sued by  banks,  and  are  payable  on  demand. 

A  hankahle  note  is  a  promissory  note  payable  at  a  bank. 

112,  Bank  discount  is  interest  paid  in  advance  for  the 
loan  of  money  on  a  note. 

1.13.  The  tenn  of  discount  is  the  time  from  the  date  of 
the  loan  to  the  maturity  of  the  note. 

a.  Bank  discount  is  payable  on  the  day  of  making  the  loan  ;  while 
legal  interest  is  payable  when  the  paper  matures.     Hence, 

b.  Bank  discount  exceeds  legal  interest,  by  interest  on  the 
legal  interest  for  the  term  of  discount. 


DISCOUNT,  317 

114:*  The  pi'oceeds  of  a  bankable  note  are  its  face,  minus 
the  bank  discount. 

US,  A  protest  is  a  written  notice  to  the  indorser  of  a 
note,  informing  him  that  the  note  has  been  presented  to  the 
maker,  at  maturity,  for  payment,  and  has  not  been  paid  ; 
and  that  the  holder  looks  to  the  indorser  for  payment  of 
the  note. 

a.  To  hold  an  indorser  responsible,  a  protest  must  he  served  on  Mm 
on  the  last  day  of  grace. 

b.  Protests  are  usually  made  out  and  served  by  a  notary  i^uhlic. 

116.  Ex.  Find  Process. 

the  proceeds  of  a  ^  mo.  3  da.z=z  Jf..l  mo. 

note     for    $2,500,      $2,500  x  .06  =  $ ISO.int. for  1  yr. 
due  in  4  months,  $i50xu.i  ^^,^1.25,  bank  discount. 

at  6^.  $2 ,500  -  $5 1 .25  z^  $2 ,i^8 .7 5,  proceeds. 

117*  In  computations  in  bank  discount, 

a.  Face  of  notez=iprincipal,  or  base; 

b.  Interest  onface^=banh  discount,  or  percentage; 

c.  Proceeds  ^^f ace— bank  discount,  or  difference. 

1~lfi    V     .    T         j  ^*  Face  X  rate  of  inter  est  zi^banJc  discount; 
*  I  IT.  Face  — bank  discount  =z proceeds. 

PROBLEMS. 

What  is  the  bank  discount  on  a  note 

1.  For  1250  due  in  3  mo.,  at  10^? 

2.  For  $81.50  due  in  6  mo.,  at  8^? 

3.  For  $1,640  due  in  15  da.,  at  6^? 
Jf.  For  $2,375  due  in  60  da.,  at  V^? 
5.  For  $495  due  in  4  mo.,  at  5^? 

Find  the  proceeds  of  bankable  notes  discounted  as  follows  : 


6.  $600  for  30  da.,  at  < 

7.  $321  for  90  da.,  at  i 


8.  $3,500  for  4  mo.,  at  6^. 

9.  $418.50forl0da.,atl0;^. 


Find  the  bank  discount,  and  the  proceeds,  of  notes  as  follows  : 


10.  $500  for  90  da.,  at  6^. 

11.  $850.31  for  60  da.,  at  5^ 


12.  $10,000  for  4  mo.,  at  7^. 

13.  $1,250  for  2  mo.,  at  8^. 


318  SUPPLEMENT, 

III.  True  Discount. 
no*  True  discount  is  the  amount  of  an  obligation  due 
at  a  future  time,  minus  the  present  worth. 

a«  Equitable  discount  is  another  name  for  true  discount. 
b.  The  equitable  2)^"es€nt  tvorth  of  an  obhgation  due  at  a 
future  time,  is  a  principal,  which,  at  a  given  rate  of  interest, 
will  amount  to  the  obligation  plus  the  discount. 

120*  In  computations  in  true  discount, 

a.  Sum  to  be  discounted  =:amounty  or  face  ; 

b.  Present  2Vorth=zbasej  or  principal ; 

c.  Amount— 2>resent  worth=true  discountj  or  percentage. 
The  process  of  finding  the  present  worth  is  therefore  the 

same  as  Case  IV,  p.  300.     Hence, 

121.  Formulas.  \  ^'/^^^'^^  wor th = amount ^{1-^ per  cent). 

(  //.  Discount  =  amount — present  worth. 
Ex.  Find  the  true  discount,  at  6^,  on  $375.70  due  in  8  mo. 

Explanation.— $375. 70,  the  sum  Process. 

due  in  8  mo.,  is  the  amount  or        S  mo.  =  .O^^per  cent ;  an  I 
face  ;  and  1.04,  the  amount  of    i  ^  ^QJ^  =  1  .o\,  amt.  of  1 ; 
1  plus  the  rate  on  1      ^S7  5 .1 0 -^  1 .0  J,  =  %S6 1.25  ; 
S'ercent      %S1 5 .1 0  ^  %S6 1.25  ^%H.U5. 

I  therefore  divide  $375.70,  the  sum  to  be  discounted,  by  1.04  or  1 
plus  the  per  cent,  and  obtain  $361.25,  the  present  worth.  I  then 
subtract  $361.25,  the  present  worth,  from  $375.70,  the  sum  to  be 
discounted,  and  obtain  $14.45,  the  required  true  discount. 

Problems. 

1.  What  is  the  true  discount  on  $100,  due  in  2  mo.,  at  6^? 

2.  On  $854  due  in  7  mo.  15  da.,  at  10^? 
S.  On  $69.50  due  in  1  yr.  6  mo.,  at  7;^? 
^.  On  $732.25  due  in  4  mo.  20  da.,  at  6^? 

'  5.  Of  $1,000  due  in  2  yr.,  at  5^^? 
What  is  the   J  6.  Of  $3,500  due  in  8  mo.  21  da.,  at  6,^? 
present  worth  I  7.  Of  $1,275  due  in  lyr.  10  mo.  12  da.,  at  10^? 
I  8.  Of  $91.42  due  in  8  mo.  20  da.,  at  6^? 


DISCOUNT,  319 

Find  the  true  discount,  and  the  present  worth, 

9.  Of  $925  due  in  1  yr.  8  mo.,  discounted  at  6^. 

10.  Of  $2,992.54  due  in  7  mo.,  discounted  at  4^^. 

11.  Of  $600  due  in  4  yr.,  discounted  at  5^. 

12.  Of  $560  due  in  1  yr.  6  mo.,  discounted  at  V^. 

Problems  in  Discount. 

1.  What  is  the  net  cash  cost  of  a  bill  of  goods  amount- 
ing to  $957.60,  on  4  months  time,  5^  off  for  cash  ? 

2.  At  6^  per  annum,  what  is  the  discount  for  the  present 
payment  of  a  note  for  $1,375.70,  due  in  9  months  ? 

3.  What  will  be  the  proceeds  of  a  note  for  $4,500,  due  in 
6  months,  discounted  at  a  bank  in  San  Francisco  ? 

4.  A  broker  buys  a  6  months  note,  at  8^  discount,  and 
pays  $2,250  for  it.     What  is  the  face  of  the  note  ? 

5.  Bought  a  bill  of  goods  amounting  to  $1,890,  on  8 
months  ;  and  cashed  it  at  a  discount  of  10^  off  for  30  days, 
and  a  further  discount  of  5^  for  cash.  What  was  the  net 
cash  cost  of  the  goods  ? 

6.  A  Western  dealer  buys  carriages  in  New  Haven,  Conn., 
amounting  to  $4,440,  on  4  months,  or  5^  off  for  cash.  How 
much  will  he  make  by  borrowing  the  money  at  a  New 
Haven  bank,  and  cashing  the  bill  ? 

7.  For  what  sum  must  I  draw  my  note,  at  4  months,  to 
borrow  $2,685  at  a  bank  in  Baltimore  ? 

8.  An  immigrant  buys  a  Kansas  farm  for  $950  ;  terms 
$500  cash,  balance  in  2  years  without  interest.  In  8  months  he 
pays  the  balance,  less  9^  discount.    How  much  does  he  pay  ? 

9.  Jan.  9  I  took  a  note  for  $344  due  in  9  months  with 
interest.  March  19a  bank  in  St.  Paul,  Minn.,  discounted 
the  note  for  me.     How  much  did  I  realize  from  the  note  ? 

10.  What  is  the  difference  between  discounting  a  bill  of 
goods  at  25^  and  10^  off,  and  discounting  the  same  bill  at 
10^  and  25^  off? 


[320] 

§  13.    COMPOUND  INTEREST. 

Interest  is  sometimes  made  payable  annually,  semi  -  an- 
nually, or  quarterly,  with  the  agreement  between  parties 
concerned  that,  if  not  paid  when  due,  it  is  to  be  added  to 
the  principal,  and  the  two  are  to  form  a  new  principal. 

122*  Compound  interest  is  the  interest  on  a  principal 
formed  by  adding  interest  to  a  former  principal.     Hence, 

I.  77ie  amount  of  the  princi2:>al  for  the  first  interest  term 
(i.  e.,  for  1  yr.,  6  mo.,  etc.)  is  the  princijml  for  the  second  in- 
terest term  ;  the  amount  of  this  principal  for  the  second  inter- 
est term  is  the  principal  for  the  third  interest  term  ;  and  so  on. 
XL  The  final  amount^  minus  the  first  principal^  is  the  com- 
pound interest  for  the  ichole  time, 

Ex.  What  is  the  compound  interest  of  $372.50  for  2 
years,  at  6^. 

Explanation.— Since  the  amount  Process. 

of  any  interest  term  is  the  prod-     i, 3  J 2, 50     Prin. 
uct  of  the  principal  multiplied  i'  OR      7  _l  * 

by  1  plus  the  rate,  I  multiply      llil^L    ^  '^  '*'^^^- 

the  principal,  $372.50,  by  1.06,      $894.85  {  }'; -^t /or 'J^."^ 
and  obtain  $394.85,  the  amount  1  06     1 -^  rate 

for    1    year.     I    multiply   tliis     '- 

amount  by  1.06,  as  before,  and     %U18.5U     Ami.  for  2  yr. 
obtain  $418.54,  the  amount  for        872.50     Prin. 
2  years.     Then  subtracting  the        <t  i  a  n  f      Ai/ 
principal,    $372.50,    from    this        ^^^'^^     ^''^' 
amount,  I  have  $46.04,  the  required  interest. 

Problems. 
At  compound  interest,  what  is  the  amount 
1,  2.  Of  $024.45  for  3  years,  at  6^  ?     At  5^  ? 
8,  Jf.  Of  $25.75  for  4  yr.  2  mo.,  at  Qfo  ?     At  4^  ? 
The  required  amount  is  the  amount  of  the  given  sum  for  4  years, 
plus  the  interest  of  this  amount  for  2  months. 

5.  Of  $856.75  for  2  yr.,  at  4^,  int.  payable  semi-annually? 

6.  Of  $364.50  for  2^  yr.,  at  Qi,  int.  payable  quarterly? 


COMPOUND   INTEREST. 


321 


What  is  the  compound  interest 

7,  8.  Of  $781  for  5  years,  at  7f,?     At  3|^? 

9.  Of  $459.26  for  3 J yr.,  at  4^  interest  payable  quarterly? 

10.  Of  $437.50  for  1|  yr.,  at  5^  interest  due  semi-annually  ? 

11.  Of  $575  for  1  year,  at  1^  a  month  ? 

i£  What  is  the  difference  between  the  simple  and  the 
compound  interest  of  $4,275  for  6  years,  at  4^? 

Savings-Baxiv  Accounts. 
123,  Savings-banks  add  to  each  depositor's  account,  at 
the  end  of  each  interest  term,  the  interest  due  on  his  de- 
posits. 

Ri  With  some  savings-banks  the  interest  term  is  6  months;  with 
some,  3  months  ;  and  with  some,  1  month. 

b.  Most  savings-banks  allow  interest  only  on  sums  that  have  been 
on  deposit  a  full  interest  term. 

c.  The  smallest  balance  on  deposit  at  any  one  time  during  an 
interest  term,  is  the  sum.  on  which  interest  is  computed  at  the 
end  of  the  term. 

Ex.  Interest  at  4^  per  annum,  payable  semi-annually  Jan. 
1  and  July  1,  how  much  was  due  Jan.  1,  1883,  on  the  fol- 
lowing account  ? 

Mechanics'  Savings  Institution  in  AccU  loith  Roht.  Williams. 
Br.  Cr. 


1882 

1881 

July 

1 

To  Cash, 

231 

50 

Nov. 

10 

By  Check, 

87 

50 

Sept. 

15 

<<       li 

05 

— 

Dec. 

1 

<<       << 

50 

— 

188^ 

1882 

March 

20 

'*  Draft, 

69 

25 

Jan. 

5 

'* 

12 

— 

May 

9 

"  Cash, 

20 

— 

April 

18 

li       i( 

u 

— 

July 

1 

"      " 

30 

75 

Aug. 

25 

"  Draft, 

no 

75 

Oct 

12 

"  Check, 

2J, 

— 

Nov. 

27 

"  Cash, 

7 

50 

1883 

Jan. 

1 

"   Check, 

8 

25 

02 


[322] 


§  14.   PARTIAL  PAYMENTS. 

I.  National  Method. 

124.  A  paHial  payment  is  a  payment  of  a  part  of  an 
obligation  that  is  due,  or  that  is  drawing  interest. 

12S.  U.  S.  Court  Rule  fob  Partial  Payments. 

I,  Mom  the  amount  of  the  principal,  computed  to  the  time 
when  the  payment  or  the  sum  of  the  payments  equals  or  ex- 
ceeds the  interest  due,  subtract  the  payment  or  the  sum  of  the 
payments, 

II.  The  remainder  is  a  new  principal,  loith  which  proceed 
as  before, 

Ex.  April  10,  1880,  a  note  was  given  for  $840,  at  6^  in- 
terest. Payments  were  made  Jan.  19,  1881,  of  $185  ;  and 
Nov.  3,  1881,  of  $30.     What  was  due  Jan.  5,  1882  ? 

Explanation.— I  first  Process. 

find  the  amount  of  xssiyr.  1  nio.  19da,  18S2  yr.  1  mo.   5  da. 

the    principal   from  ^        ^^ 

the  date  of  tlie  note I 

to  Jan.  19,  1881,  the  9  mo.   9  da.  11  mo.  16  da. 

'^::SS:^.  ^  -•  ^ ^- =^-^ -= .^^^;,;>- -^. 

Subtracting  $185,  1^:0/^65  =  1.0J,65,  amt.  of  1. 

the  payment,  from       %S40  x  1.0465  =  %879.06,  amt. 

this  amount,  I  ob-     %879. 06  -  $185  =  $694. 06,  Tieic  pnn. 

*tain  a  remainder       11  mo.  16  da.  =  11.5^  7no.=  .057^,  per  cent. 

of  $694.06   for  a  l+.057i  =  1.057lamt.of  1. 

new   principal.      ^^94.06  x  1.057^  =  $734.08,  amt. 

$30,  the  second  ^734,08- $30  -  $704.08,  hal.  due. 

payment,   made 

Nov.  3, 1881,  did 

not  exceed  the  interest  due  ;  and  I  next  find  the  amount  of 

$694.06,  from  Jan.  19,  1881,  to  Jan.  5,  1882,  which  is  $734.08. 

Subtractini^  from  this  amount  $30,  the  payment  made  Nov.  3^ 

lySl,  I  have  $704.08,  the  required  amount  due  Jan.  5, 1882. 


PARTIAL    PAY3fENTS, 


323 


Condensed  Form  of  Written  Work. 


Dates. 
)  yr.  4  mo.  10  da. 


19 
5 


Interest  Periods. 


9mo.  9da.=  9.3  mo. 
11       16       =11.  Ci  mo. 


1.0465 
1.0571 


Principals. 

$840 
694.06 
704.08 


$879.06 
734.08 


Pay'ts. 


$185 
30 


Problems. 

1.  A  store  was  sold  in  Indianapolis,  Ind.,  for  $4,750; 
payments,  $2,100  cash,  $1,200  in  one  year,  and  the  balance 
in  two  years.     How  much  was  the  last  payment  ? 

2,  Memorandum: — Nov.  26,  1880,  gave  a  note  for  $1,780, 
at  6^.  June  25,  1881,  paid  $160  ;  Nov.  1,  1881,  paid  $525. 
How  much  was  due  March  11,  1882  ? 

S.  On  a  note  for  $450,  dated  Louisville,  Ky.,  April  15, 
1881,  $200  was  paid  Jan.  24,  1882.  What  amount  was  due 
Nov.  8,  1882? 

^.  A  mortgage  for  $12,500,  dated  Detroit,  Mich.,  Oct.  31, 
1879,  bears  the  following 

Indorsements :— May  28,  1880,  $900;   Jan.  1,  1881,  : 
July  14, 1881,  $375;  Nov.  29,  1881,  $745 ;  Aug.  11,1882, 

What  amount  was  due  Jan.  22,  1883  ? 

B,     S765  Springfield,  Mass.,  March.  lU^  1880. 

(y/i  c/e7nand,   Qy    ht07?7Me  ^o  ha7/  io   Q/noniad   077ietdon,  oi  otc^et, 
(Reverb   c^ctnc/ied  ^ixi7j=/tve  Q^JoCuztd,  wUn  tntete^t,  Joi  value  te-^ 

Indorsements :— Oct.  31,  1881,  $50;  June  11,  1882,  $285. 
Find  the  balance  due,  Sept.  25,  1882. 

6>      ^j>  ^QQ  Washington,  D.C.,  Btc.  5, 1881. 

^Tie  'ueaz  a/let  e/ale,  ti/e  hionif'de  lo  ha7/  ^o  ^^^otae  Q2).  'crai^?net, 

/ot  va/ue  tecetvec/.  Mo^tk^cm,    ^/ati    ^  ^o. 

Indorsements: — July  2,  1882,  $350;  Nov.  8,  1882,  $675. 
W^hat  was  the  balance  due  March  18,  1883? 


324  SUPPLEMENT. 

126.  When  settlements  are  made  within  a  year  after 
interest  commences,  the  computations  of  interest  are  often 
made  by  the 

II.  Mercantile  Rule  for  Partial  Payments. 

I.  Compute  the  interest  on  the  principal  for  the  lohole  time, 

II.  Coynpute  the  interest  on  each  payment,  from  its  date 
to  the  time  of  settleme^it, 

III.  Subtract  tJie  amount  of  the  payments  from  tJie  amount 
of  the  principal. 

In  making  computations  by  this  rule,  exact  interest  for  daj^s  is 
commonly  considered. 

Problems. 

1.  If  I  borrow  $1,250  for  1  yr.,  at  6^^,  and  pay  $625  in 
5  mo.,  how  much  do  I  owe  at  the  end  of  the  year? 

2.  On  a  mortgage  for  $3,125,  dated  July  5, 1880,  a  payment 
of  $1,450  was  made  April  23,  1881.  What  amount  was  due 
Jan.  17, 1882? 

S.  June  7,  1881, 1  borrowed  $8,000,  at  4-J^.  Jan.  28, 1882, 
I  paid  $3,500;  and  Aug.  14,  1882,  $2,750.  How  much  was 
due  Feb.  3,  1883? 

III.  Annual  Interest. 

127*  In  some  States,  a  written  obligation  containing  the  words 
"  with  interest  annually,"  or  "  with  annual  interest,"  is  a  legal 
contract  on  the  part  of  the  maker  of  the  obligation,  to  pay  sim- 
ple interest  on  the  principal  at  the  end  of  each  year,  whether  the 
principal,  or  any  part  of  it,  is  due  or  not.  If  the  maker  fails  to 
pay  the  interest  at  the  end  of  each  year,  the  law  allows  to  the 
holder,  in  the  nature  of  damages,  simple  interest  on  the  unpaid 
yearly  interests,  until  they  are  paid ;  but  it  does  not  allow  inter- 
est on  these  interests. 

128.  Annual  interest  is  simple  interest  upon  the  prin- 
cipal, and  upon  each  year's  interest  of  the  principal,  due  and 
unpaid. 


PARTIAL  PAYMENTS.  325 

120 •  Ex.  1.  A  note  for  $850,  with  interest  annually,  was 
taken  up  at  the  end  of  5  years.  How  much  interest  had 
accrued  ? 

Full  Solution. 
Interest  due  on  principal  at  the  end  of  each  year^  $850  x  .06z=$-51 
1  yr.^s  int.  draws  simple  int.  for  Jf  yr.  -f  3  yr.-\-2  yr.  -{-lyr.^iz  10  yr. 
10  yr.at  6fc  —  10x.06  =  .Q>0  — per  cent. 
Simple  int.  on  1  ijr.h  int.  for  10  yr.,  $51  x  .60  =  $  80.60 
Interest  on  p)Tincipal  for  5  yr.,  $850  x  .06  x  5=z  255 

Total  interest,  $285.60 
Ex.  2.  What  is  the  interest  of  $350  for  4  yr.  8  mo.  12  da., 
interest  payable  annually  ? 

Full  Solution. 

Interest  due  on  principal  at  the  end  of  each  year,  $350  x  .06:=$21 

1  yr.^s  int.  draics  simple  int.  for  3  yr.  8  mo.  12  da.-\-2  yr.  8  mo. 

12  da.  +  l  yr.  8  mo.  12  da.-\-8  mo.  12  da.=8  yr.  9  mo.  18  da, 

8  yr.  9  mo.  18  da.  at  6fo  =  105.6  mo.  =  .528,  per  ciiit. 
Simple  int.  on  1  yr.^s  int.  for  8  yr.  9  mo.  18  da., 

$21 X. 528 =  $11.09 

Jf  yr.  8  mo.  12  da.  =56. 4-  mo.  =  .282, per  cent. 
Int.  on  prin.for  4  yr.  8  mo.  12  da.,  $350  x  .282=   $98.70 

Total  interest,  $109.79 
Hence,  when  interest  is  payable  annually,  the  total  interest 
of  any  sum  of  money  for  any  given  time  is  made  up  of 

I.  The  interest  on  the  principal  for  t/ie  ivhole  time  ; 

II.  The  simple  interest  on  one  year'^s  int&i^est  for  the  sum 

of  the  periods  of  time  the  several  yearly  interests  remain  U7i- 

paid. 

Problems. 

At  6^,  interest  payable  annually,  find  the  interest  of 


1.  $800  for  5  years. 

2.  $241  for  4  yr.  2  mo. 

3.  $124  fov  6  yr.  8  mo. 


If.  $172  for  3  yr.  3  mo.  3  da. 

5.  $387.50  for  5  yr.  4  mo.  15  da. 

6.  $574.45  for  3  yr.  9  mo.  14  da. 


7.  $96.84  from  Nov.  27,  1880,  to  July  10,  1884. 

8.  $1,000.40  from  April  1,  1880,  to  July  10,  1885. 


326  SUPPLEMENT. 

At  6^,  interest  payable  annually,  what  is  the  amount 

9,  Of  a  note  for  $600,  which  has  run  4  yr.  3  mo.  15  da.  ? 

10.  Of  a  mortgage  of  $2,000,  for  2  yr.  7  mo.  20  da.  ? 

11.  Of  $520  from  Oct.  10,  1882,  to  the  present  day? 

IV.  Special  State  Law^s  for  Partial  Payments. 
ISO*  Vermont. 
When  notes,  bills,  or  other  obligations  draw  annual  interest, 
and  any  part  or  the  whole  of  this  interest  remains  unpaid, — 

/.  Payments  draw  interest  to  the  end  of  the  yearly  interest  terms 
in  which  they  are  made. 

II.  The  amount  of  any  payment  or  payments  made  in  any  in- 
terest terms  is  applied 

1st. — To  cancel  interest  due  on  unpaid  yearly  interests  ; 
2d. —  To  cancel  unpaid  yearly  interests  ; 
3d. —  To  cancel  the  principal. 
The  last  balance  must  be  computed  to  the  date  of  settlement. 

13T*  New  Hampshire. 
Payments  not  exceeding  interest  due  at  the  end  of  the  year.,  and 
made  expressly  on  account  of  interest  accruing  hut  not  yet  due,  do 
not  draw  interest.     At  the  end  of  the  year  they  must  be  applied  to 
the  payment  of  the  interest  then  accrued. 

In  all  other  respects  the  law  is  the  same  as  in  Vermont. 
132,  Connecticut. 
/.   W/fcn  a  year'^s  interest  or  more  has  accrued  at  the  time  of  a 
payment,  or  when  any  payment  is  less  than  the  interest  due,  and 
also  in  case  of  the  last  payment,  the  interest  is  computed  by  the 
U.  S.  Court  Rule. 

II.  When  less  than  a  year'^s  interest  has  accrued  at  the  time  of 
any  payment,  except  the  last,  the  amount  of  the  payment  from  its 
date  to  the  end  of  the  full  year  is  deducted  from  the  amount  of  the 
principal  for  the  full  year  ;  the  remainder  is  the  7ietv  principal 
for  the  next  interest  tenn. 

Note. — Pupils  in  any  of  these  three  States  should  be  required  to 
solve  all  the  problems  in  partial  payments  (page  323)  by  the 
special  law  for  their  State. 


[327] 

§  15.  BONDS. 
133.  A  bonil  is  a  written   obligation  from  one  party, 
securing  to  another  the  payment  of  a  given  sum,  at  or  be- 
fore a  specified  time,  with  interest  payable  annually,  semi- 
annually, or  quarterly. 

a.  Bonds  are  issued  for  the  purpose  of  borrowing  money. 

b.  The  principal  bonds  bought  and  sold  by  brokers  are  govern- 
ment, state,  city,  and  railroad  bonds. 

United  States  securities  or  government  bonds  are  desig- 
nated as  registei'ed  bonds  and  coupon  bonds. 

134:.  A  registered  bond  is  one  that  is  recorded  on  the 
books  of  tbe  Treasury  Department  at  Washington,  as  the 
property  of  a  certain  person. 

13S*  A  coupon  is  an  interest  certificate  attached  to  a 
bond. 

136.  A  coupon  bond  is  a  bond  to  Avhich  coupons  are 
attached. 

"When  the  interest  on  a  coupon  bond  is  paid,  a  coupon  is  detached, 
by  the  holder,  and  given  up  as  a  receipt. 

Bonds  are  usually  named  from  the  authority  that  issued 
them  and  the  rate  of  interest  they  bear. 

Virginia  6's  are  bonds  bearing  6^  interest,  issued  by  the  State  of 
Virginia. 
Brokerage  is  ^^  to  |^  of  the  par  value  of  stocks  and  bonds. 

List  of  the  Principal  U.  S.  Bonds  outstanding  Jan.  1, 1882. 


Names  of  Bonds, 

When  Redeemable. 

Rates  of 
Interest. 

Interest  Payable. 

Four-and-a-halfs  of  1891 

After  Sept.  1, 1891. 

'iifo 

Quarterly. 

Fours  of  1907 

After  July  1, 1907. 

H 

Quarterly. 

Three-and-a-halfs 

At  option  of  Gov't. 

3if» 

U.  S.  Pacific  KR.  cur-  ) 
rency  sixes                   ) 

j  July  1, 1892,  and  ) 
(      July  2, 1894.     f 

6^ 

Semi-annually. 

Currency  sixes 

1895  to  1899. 

6^ 

Semi-annually. 

328  S  UPPL  EM  EN  T. 


137*  In  compu- 
tations in  bonds 


IS 
CO 


a«  Par  ?;a/i«e  or  face  of  bond  —  tetf  ; 

b.  Rate  of  premium  or  discount  z=i  per  cent ; 

c.  Market  value  =  amount  or  difference. 
Hence, 

/.  Market  xalue=fa^  of  bond  x\^'^_  '.^^  %^^Zur!!^: '''' 

„  ^       ^,      -,  ,  .     ,  i\-\- rate  of  premium,  or 

II.  Fa^  ofbona  =  marlc  t «'«'««  H- ]  j  _  ^^^  ofdmount. 

III.  Rate  of  premium  =  (market  valite  —face)  -r-  face. 

IV.  Rate  of  discount  =  (face  —  market  value)  -^face.  ' 

Problems. 
Find  the  market  value  of  the  following  securities: 

1.  Of  Panama  RR.  500-dollar  bonds,  at  67,  brokerage  ^^. 

2.  Of  Government  4^8  of  1907,  at  118|,  brokerage  -J^. 
How  many  100-dollar  bonds  can  be  bought 

3.  For  $21,535,  of  Tenn.  5's,  at  91,  brokerage  J^  ? 

Jf,  For  $3,944,  of  K  J.  Central  RR,  at  115f,  brokerage 


li^ 


9 


For  $21,540,  of  Central  Pacific  RR,  at  89f,  brokerage 

6.  For  $7,048,  of  Mich.  7's,  at  110,  brokerage  ^fo  ? 
Which  is  the  better  investment,  and  how  much  the  better, — 

7.  Michigan  7's  of  '90,  at  115;  or  Missouri  6's,  at  112  ? 

8.  Georgia  sevens,  at  119;  or  Virginia  sixes,  at  116  ? 
P.  5^  bonds,  at  102  ;  or  S^fo  bonds,  at  87|  ? 

10.  Government  fours,  at  102  ;  or  four-and-a-half s,  at  113  ? 

11.  How  much  will  3  1,000-dollar  Government  four-and-a- 
half  s  cost,  at  S^fo  premium  ? 

12.  A  capitalist  invested  $20,352  in  New  York  City  bonds, 
at  4^  discount.     What  amount  in  b®nds  did  he  receive  ? 

13.  Currency  6's  at  105,  will  pay  what  ^  on  investment? 
14'  I  invest  $20,500  in  Virginia  sixes,  at  111.    The  annua] 

interest  is  what  per  cent  on  the  investment? 


[329] 

§  16.  EQUATION  OF  PAYMENTS. 
139 •  Equation  of  payments  is  the  process  of  finding 
the  time  for  paying,  in  one  sum,  several  debt3  due  at  differ- 
ent times,  without  loss  to  debtor  or  creditor. 

a.  The  ter^n  of  credit  is  the  time  that  must  elapse  be- 
fore a  debt  matures. 

b.  The  equated  time  is  the  time  for  paying,  in  one  sum, 
several  debts  due  at  different  times. 

14z0,  Ex.  Find  the  equated  time  for  the  payment  of  $300 
due  in  2  mo.,  $500  due  in  4  mo.,  and  $200  due  in  6  mo. 

Full  Solution. 

Int.  of   $800  for  2  mo.  =  int.  of  $1  for  300  x  2  mo.,  or    600  mo, 

"     "      500    '*  4   "    ==    ''    "    1    "   SOO  X  4-  ''    "  2,000   " 

"     "      200    ''  6   ''    —    "    "    1    "   200  X  ^   "     "  1,200   " 

"     ''$lfidb    ''   ?    ''    rr    "    "    i    ''  1  thousandth  of  3,800    " 

8,800  mo.  -r- 1^000  =i  3.8  mo.  =  3  mo.  2 J/,  da.,  equated  time.  Hence, 

Rule. 
Multiply  each  tenn  of  credit  by  the  numher  expressing  the 
payment;  and  divide  the  sum  of  the  products  by  the  number 
expressing  the  sum  of  the  pay7nents. 

The  result  will  be  in  the  same  denomination  of  time  as  the  given 
terms  of  credit. 

Problems. 

1.  Find  the  equated  time  for  the  payment  of  $200  due  in 
3  mo.,  $240  due  in  ^^  mo.,  and  $320  due  in  5  mo. 

2.  I  owe  $425  due  to-day,  $150  due  in  6  months,  and  $275 
due  in  9  months,  and  I  wish  to  give  one  note  due  at  the 
equated  time.  On  w^hat  date  must  the  note  be  made  pay- 
able? 

3.  Jan.  1,  a  merchant  owes  the  sums  $  700  due  Mar.  15. 
named  in  the  margin,  and  he  gives  his  250  "  Apr.  1. 
note  for  the  entire  amount  payable  in  825  "  Apr.  15. 
one  sum,  at  the  equated  time.  When  1,000  "  May  10. 
does  the  note  mature  ?                                    1,425    "    June   1. 


[330] 

§  17.  EXCHANGE. 
141.  Exchange  is  a  transaction  in  wliich  a  party  in 
one  place  pays  money  to  a  party  in  another  place,  by  an 
order  upon  a  third  party,  and  without  the  transmission  of 
money. 

a.  Domestic  or  inland  exchange  relates  to  remittances 
made  between  places  in  the  same  country. 

b.  Foreign  exchange  relates  to  remittances  made  between 
places  in  different  countries. 

14i2*  A  draft  or  hill  of  exchange  is  a  written  order  for 
money,  drawn  in  one  place  and  payable  in  another. 

Example. — A  party  in  Cleveland,  -wishing  to  pay  a  creditor  in 
New  York,  buys  at  a  Cleveland  bank  a  draft  on  a  New  York 
bank,  payable  to  the  order  of  the  party  in  New  York.  The 
Cleveland  party  sends  this  draft  to  his  creditor  in  New  York, 
and  the  latter  indorses  it,  presents  it  to  the  New  York  bank,  and 
receives  the  face  of  the  draft  in  money. 

14:3.  A  sight  draft  is  a  draft  payable  at  sight,  i,  e.,  when 
presented. 

14:4:.  A  time  draft  is  a  draft  payable  at  a  future  time 
named  in  it. 

a.  Grace  is  allowed  on  time  drafts,  but  not  on  sight  drafts. 

b.  Time  drafts  are  subject  to  bank  discount  for  the  term  of  credit 
given. 

C.  Any  creditor  may  give  a  draft,  or  draw  on  a  debtor. 
Note. — See  forms  of  drafts,  page  3G2. 

14S.  The  parties  to  a  transaction  in  exchange  are  the 
drawer  or  iiiaker,  the  buyer  or  remitter^  the  drawee,  and 
\hiQ  payee, 

1.  The  drawer  or  payer  is  the  party  who  signs  or  issues 
the  draft. 

2.  The  buyer  or  remitter  is  the  party  who  purchases  the 
draft. 

S,  The  drawee  is  the  party  pn  whom  the  draft  is  drawn. 


EXCHANGE,  331 

4.  The  payee  is  the  party  to  whose  order  the  draft  is  made 
payable. 

The  maker  and  remitter,  or  the  remitter  and  payee,  may  be  the 
same  party ;  in  either  of  which  cases  there  will  be  but  three 
parties  to  the  transaction. 

14:6.  An  acceptance  is  a  w^'itten  promise  of  the  drawee 
to  pay  the  draft  at  maturity. 

A  drawee  accepts  a  draft,  by  writing  across  its  face  Accepted,  fol- 
lowed by  his  name.  This  acceptance  makes  him  responsible  for 
the  payment  of  the  draft  at  maturity. 

14:7 •  The  balance  of  trade  between  places  or  countries 
is  the  difference  between  the  amounts  due  to  and  from  each 
place  or  country  by  the  other. 

Example.— St.  Louis  owes  New  York  $12,375,090,  and  New  York 
owes  St.  Louis  $10,500,000.  The  balance  of  trade  between  the 
two  cities  is  $1,825,000  against  St.  Louis  and  in  favor  of  New 
York.  In  this  case,  in  St.  Louis,  exchange  on  New  York  is  at  a 
premium ;  and  in  New  York,  exchange  on  St.  Louis  is  at  a  dis- 
count. 

Drafts  are  bought  at  par,  at  a  premium^  or  at  a  discount. 

148.  Bate  of  exchange  is  the  difference  between  the 
face  of  a  draft  and  its  cost. 

The  rate  of  exchange  is  affected  by  the  condition  of  the  balance 
of  trade  between  the  two  places  concerned. 

In  computations  in  exchange 


a.  Face  of  draft = base. 

b.  Mate  of  exchange  ^=  per  cent. 

c.  Premium  or  discount  =1  per- 

centage. 


d.  Cost  of  draft  =  < 


Amount  or 
difference. 
e.  JF'ace  minus  bank  discount  = 
proceeds  of  time  draft. 


Hence,  a7i7/  prohlem  in  exchange  comes  under  one  or  more 
of  the  cases  in  percentage. 

\   I.  Face  x\{  +  '■"''  "f  ^«'«»'«"^  «'•  I  =  Cost. 
jL4:U.         J  (  i  —  7'ate  of  discount,         ) 

Formulas,     t  ^^    ^^^^  -^  -j  ^  +  ^''^^^  ^^'  premiuin,  or  )  ^^^^^^ 
^     *  '    \l  —  rate  of  discount,         ) 


332  SUPPLEMENT. 


Problems. 
Find  the  cost  of  a  sight  draft 


Jf.  For  $631,  at  2^^  discount. 

5.  For  6389,  at  15^  discount. 

6.  For  §2, 750,  at  l^fo  discount. 


1.  For  $250,  at  ^fo  premium. 

2.  For  $456.50,  at  2|^  premium. 

3.  For  $1,325  at  15^  premium. 
Find  the  face  of  a  sight  draft  which  cost 

7.  $116.44,  at  Iki  premium.    I      9,  $835.63,  at  ^^o  discount. 

8,  $1,000,  at  i^  premium.        1    10,  $3,521,  at  2|;^  discount. 
Find  the  cost  of  a  draft 

11,  For  $500,  at  60  da.,  premium  at  -J^^,  interest  4^. 

12,  For  $1,732,  at  30  da.,  premium  ^^  interest  6^. 

13,  For  $2,000,  at  10  da.,  discount  i^,  interest  1^  per  mo. 
H.  How  much  will  it  cost  me  to  make  a  remittance  of 

$1,750  from  Boston  to  Charleston,  exchange  on  Charleston 
being  at  98|^  {i.  e.,  at  1|^^  discount)  ? 

15,  How  much  must  a  man  in  St.  Louis  pay  for  a  sight 
draft  on  Boston  for  $532,  exchange  being  102^^? 

16,  A  merchant  in  Burlington,  Iowa,  wishing  to  remit  to 
a  creditor  in  Philadelphia,  buys  a  draft  at  |^  premium,  and 
pays  $2,460  for  it.     What  is  the  face  of  the  draft  ? 

17,  A  lawyer  in  Wheeling,  W.  Va.,  having  $745.50  be- 
longing to  a  client  in  Denver,  purchases  a  sight  draft  with 
it  on  a  Denver  broker,  at  103.  What  is  the  face  of  the  draft? 

18,  What  must  I  pay  in  Cincinnati  for  a  draft  on  New 
York  for  $2,400,  payable  60  days  after  sight,  exchange  on 
New  York  being  100|? 

On  a  time  drafts  compute  both  the  discount  and  the  rate  of 
exchange,  on  the  face  of  the  draft. 

19,  A  San  Francisco  banker  sold  a  draft  on  New  York  for 
$6,117,  payable  30  days  after  date,  when  exchange  on  New 
York  was  at  3^  premium.    What  was  the  face  of  the  draft  ? 

20,  A  broker  in  Boston  pays  $1,352.62  for  a  draft  payable 
at  Columbus,  Ohio,  30  days  after  sight,  at  98|^.  What  is  the 
face  of  the  draft  ? 


[333] 

§  18.  RATIO  AND  PROPORTION. 
I.  Ratio. 
ISO,  Ratio  is  the  relative  value  of  two  like  numbers, 
expressed  by  the  quotient  of  the  first  divided  by  the  second. 

The  ratio  of  13  to  4  is  3  (12-^4=3);  of  $34  to  $6  is  4. 
Ratio  is  always  expressed  by  an  abstract  number. 

1^1.  The  terms  of  a  ratio  are  the  numbers  whose  val- 
ues are  compared. 

a.  The  antecedent  is  the  first  term. 

b.  Tlie  consequent  is  the  second  term. 

C.  In  expressing  the  ratio  of  13  to  4,  13  and  4  are  the  terms  of  the 

ratio,  13  is  the  antecedent,  and  4  is  the  consequent, 
d.  The  terms  of  a  ratio  are  called  a  couplet, 

ltj2»  The  sign  of  ratio  is  the  colon  (:),  written  between 
the  terms.     It  is  read  "the  ratio  of." 
13  :  4  is  read  ''the  ratio  of  13  to  4." 

Ratio  may  also  be  expressed  in  the  form  of  a  fraction,  the  antece- 
dent being  written  for  the  numerator,  and  the  consequent  for  the 
denominator.     Thus,  13:4  =  ^^-. 

Exercises. 
Read  Express,  in  both  forms,  the  ratio 


11.  Of  $.48  to  $.00. 
X^.  Of  $9  to  $15. 
13.  Of  28  yd.  to  4  yd. 
U.  Of  5  lb.  to  621  lb. 


i.  32  :  8=134.  I  2.  ^=,4..  \  7.  Of  12  to  4. 
5.  56  men:  7  men  =  8.  j  ^.  Of  49  to  7. 
^.     7  men  :  56  men=r|.   .     ^  Of  5  to  30. 

5.  40  bn.    :  16  bu.rr:2|-.  |  10.  Of  9  to  72. 

6.  16  bu.    :  40  bn.r^f.     |  15.  Of  1,756.25  mi.  to  28.1  mi. 

IS 3*  A  simple  ratio  is  a  ratio  that  has  but  one  ante- 
cedent and  one  consequent. 

1S4»  A  compound  ratio  is  the  ratio  of  the  products  of 
the  like  terms  of  two  or  more  simple  ratios. 
The  simple  ratio  of    16  :  4  is  ^£-,  or  4; 
*'      ''       9:3"§,  or3; 

''compound  ''      "  -j    g  !  3  |-  is  i;|- x §,  or  4 x 3,  or  13. 


334 


SUPPLEMENT. 


155.  Rules  for  Finding  Ratios. 
I.  To  find  a  simple  ratio  : — Dimde  the  antecedent  by  the 
consequent. 

II.  To  find  a  compound  ratio  : — Dimde  the  product  of  the 
antecedents  of  all  the  simple  ratios  by  the  product  of  all  the 
consequents. 

If  the  terms  of  a  ratio  are  denominate  numbers,  reduce  them  to  the 


same  denomination. 


Find  the  ratio 


Problems. 


1.  Of  45  to  15.      Jf,  Of  12  h.  to  16  li. 

2.  Of  14  to  112.   5,  Of  $1.25  to  $6.25. 

3.  Of  29  to  8.       6.  Of  iVi  yd.  to  2^  yd, 
Find  the  unknown  number  in  each  of  the  following  ex 

pressions : 

ConseqiieDtg. 


7,  3.5  mo. :  24.5  mo. 

8,  35  da. :  4  da.  9  h. 

9,  1  gall  qt. :  10  gal. 


Antecedent. 

10.  1,275 

11.  $300 

12.  —  rd. 


Consequent. 
425 
$- 
5  ft. 


Ratio. 

'  Antecedei 

.75 

13  r 

5: 

66 

.       30: 

Ratio. 


=  90 


Which  is  the  greater,  the  ratio 

U,  Of  16  to  24,  or  the  ratio  of  28  to  84  ? 

15,  Of  4  ft.  to  3  yd.,  or  the  ratio  of  96  rd.  to  1  mi.  ? 

16,  The  antecedents  of  a  compound  ratio  are  12,  4,  and 
8;  and  the  consequents  are  36,  11,  and  21.  What  is  the 
compound  ratio  ? 

17,  Compound  the  ratios  5  :  8,  72  :  3,  30  :  5. 

18,  The  antecedents  are  ^\,  f ,  and  f  ;  and  the  consequents 
are  |,  6,  and  |^.     What  is  the  compound  ratio  ? 

II.  Simple  Proportion. 
156.  Proportion  is  an  equality  of  ratios. 

157*  Simple  proportion  is  an  equality  of  two  simple 
ratios. 

Since  the  ratio  of  15  to  3  equals  the  ratio  of  10  to  2,  these  two 
ratios  form  the  proportion,  15  :  3:  :10  :  2. 


RATIO   AND   PROPORTIOK  335 

138 •  The  si^n  of  proportion  is  the  double  coloii  (::), 
written  between  the  ratios.     It  is  read  "  as  "  or  "  equals." 

a.  The  proportion  15 :  3:  :10 :  2  is  read  *'  15  is  to  3  as  10  is  to  2,'* 
or  "the  ratio  of  15  to  3  equals  the  ratio  of  10  to  2." 

b.  Proportion  may  also  be  expressed  by  the  sign  of  equality. 
Thus,  15:3=10:2. 

Id9»  Four  numbers  are  required  to  express  a  simple 
proportion. 

a.  The  extretnes  of  a  proportion  are  the  first  and  fourth 

terms. 
h*  The  means  of  a  proportion  are  the  second  and  third  terms. 

Writing  the  proportion  12 : 4:  :15 : 5  in  the  fractional  form,  and  re- 
ducing the  expressions  to  similar  fractions — keeping  the  factors 
separate — we  have 

12 . .  15_12X5  . .  15X4 
4**5        4X5* '   5X4 

The  factors  12  and  5  of  the  first  numerator  are  the  extremes  of  the 
given  proportion;  the  factors  15  and  4  of  the  second  numerator 
are  the  means ;  and  the  products  of  these  two  sets  of  factors  are 
equal.    Hence, 

160,  Peinciple. — The  product  of  the  extremes  equals  the 
prodiict  of  the  means. 

Applying  General  Problems  XIV,  XV,  XVI,  page  104, 
to  this  Principle,  we  have  the  following 

f    _    Extreme  X  extreme        .        . 

I     I.  ■— — — =  the  other  mean. 

161.  Formulas.  J  ''*^''-  ™^«« 

j  ^^    Mean  X  mean      ^,        . 

II.  -: =  the  oilier  extreme. 

L        either  extreme 

The  solution  of  a  proportion  is  the  process  of  finding  one 
of  its  terms,  when  the  other  three  are  given. 

Problems. 
Find  the  unknown  term  in  each  of  these  20  proportions : 


1.  6:  30::  7:  -. 

2.  1:    3::4:  — . 


^.6:    8::  —  :12. 
^.  4:  -::    3:   9. 


5.  —  :  10::  18:  5. 

6.  9:  —  ::    3:5. 


336 


SUPPLEMENT. 


7. 

'12 

9 

8, 

18 

24 

9. 

5 

9 

10. 

10 

— 

11. 

— 

21 

12. 

40 

— 

13. 

12.5 

5 

16 

— 

24 

— 

— 

8. 

21 

14. 

35 

15. 

8 

5.3. 

27 

"~   1 

11 

lOi 

8f  : 

:   — : 

7. 

15. 

— 

A  : 

:   A: 

A. 

16. 

$.16 

$.48  : 

:-qt.: 

3  qt. 

17. 

10  lb. 

—lb.  : 

:   15: 

6. 

18. 

— oz. 

12  oz.  : 

:   $4: 

$9. 

19. 

21 

—  : 

:  9  yd. 

3  yd. 

20. 

12mi. : 

2  mi.  : 

:   19: 

— 

162.  From  the  preceding  problems  it  will  be  seen  that 
The  first  term  of  a  proportion  is  greater  or  less  than  the  second^ 
according  as  the  third  term  is  greater  or  less  than  the  fourth. 

103*  In  any  problem  in  simple  proportion  three  terms 
are  given,  and  a  fourth  term  is  required. 

a.  Two  of  the  three  given  terms  are  like  numbers;  and  the  ratio 
of  these  two  terms  equals  the  ratio  of  the  other  known  term  and 
the  required  term. 

b.  To  tnake  a  statement  in  proportion,  is  to  arrange  the  three 
given  terms,  so  that  two  of  them  form  one  ratio ;  the  remaining 
term  and  the  required  term,  another  ratio ;  and  the  ratios,  a  pro- 
portion. 

Ex.  If  16  acres  of  land  produce  424  bushels  of  wheat,  27 
acres  will  produce  how  many  bushels  ? 


Process. 
16  A.:27  A.  ::  42i  bu.:-bu, 
S3 

27  X  0^  hu.       HSlhu. 


x& 


715^  hu. 


Explanation.  — 16 
acres  aud  37  acres 
form  the  first  ratio, 
and  424  bushels 
and  —  bushels 
form  the  second 
ratio.  Since  16 
acres  are  less  than 

27  acres,  424  bushels — produced  from  16  acres — must  be  less 
than  the  number  of  bushels  produced  from  27  acres.  I  there- 
fore write  424  bushels  for  the  first  term  of  the  second  ratio,  or 
the  third  term  of  the  proportion;  and,  since  this  term  is  less 
than  the  required  term,  I  write  16  acres — the  less  of  the  two 
given  like  numbers — for  the  antecedent  of  the  first  ratio,  and 
27  acres  for  the  consequent. 

Solving  the  proportion,  I  have  715J  bushels,  the  required  term. 


RATIO   AND  PROPORTION.  337 

If  424  bushels  are  made  tlie  consequent  of  the  second  ratio,  16 
acres  must  be  the  consequent  of  the  first  ratio,  and  the  required 
number  of  bushels  will  be  the  third  term.     Thus, 
27A.:16A.::— bu.:424bu. 

16 4z.  Rule  for  Simple  Peoportioit. 
I.  For  the  third  term  :  —  Write  the  number  that  is  of  the 
same  kind  as  the  required  term. 

II.  For  the  first  ratio  :  —  Write  the  other  tioo  numbers,  the 
greater  for  the  second  term,  when  the  required  term  is  to  be 
greater  than  the  third  term;  and  the  less  for  the  second  term, 
when  the  required  term  is  to  be  less  than  the  third  term, 
III.  To  find  the  fourth  or  required  term : — 
Multiply  the  second  and  third  terms  together,  and  divide 
the  product  by  the  first  term. 

a.  When  necessary,  reduce  the  terms  of  the  first  ratio  to  the 
same  denomination,  before  solving  the  proportion. 

b.  Cancel  all  like  factors  from  the  given  extreme  and  either 
m.ean. 

Problems. 
Find  the  cost 

1.  Of  32  bu.  of  lime,  at  $14.25  for  57  bu. 

2.  Of  9i  lb.  of  beef,  at  1 1.7 8^  for  12f  lb. 

S.  Of  11  yd.  of  broadcloth,  when  6  yd.  cost  130.75. 
Jf..  Of  a  barrel  of  flour,  when  23  pounds  cost  $1.15. 
5.  Of  27  rm.  6  quires  of  paper,  at  $13.87^  for  11  rm.  2 
quires. 

e.  Of  9f  cd.  of  wood,  if  15.75  cd.  cost  $94.50. 

7.  Of  7  centals  of  wheat,  @  $1.15  a  bushel. 

8.  Of  3.25  tons  of  hay,  when  2.9  tons  cost  $24.65. 

9.  Of  44|  gal.  of  milk,  if  53  gal.  3  qt.  cost  $2.68. 

10.  If  a  carriage  wheel  makes  802  revolutions  in  running 
3  mi.  60  rd.  12  ft.,  how  many  revolutions  will  it  make  in  run- 
ning 25  mi.  15  rd.  12  ft.  ? 

11.  If  475  yards  of  sheetings  are  made  from  209  pounds 
of  cotton,  how  many  yards  of  sheetings  can  be  made  from 
645  pounds  of  cotton  ? 

P 


338  SUPPLEMENT. 

12,  If  an  ocean  steamer  runs  2,337  mi.  in  10  da.  6  h.,  how 
many  miles  will  she  run  in  5  da.  9  h.  ? 

13.  The  interest  on  a  certain  sum  of  money  for  4  mo. 
13  da.  is  $139.59.  What  is  the  interest  on  the  same  sum  for 
5  mo.  18  da.? 

IJf.  How  high  is  a  church  spire  whose  shadow  is  119  ft. 
long,  when  a  flag-staff  45  ft.  high  casts  a  shadow  56 J  ft.  long  ? 

15.  If  there  are  16  lb.  of  oxygen  in  18  lb.  of  water,  how 
much  oxygen  is  there  in  75  lb.  of  water  ? 

16.  If  2\\  qt.  of  milk  are  required  for  5|  lb.  of  cheese, 
how  many  quarts  will  be  required  for  10.5  lb.  ? 

17.  What  is  the  cost  of  J  of  an  acre  of  land,  at  the  rate 
of  $187.50  for  -J  of  an  acre  ? 

18.  The  liabilities  of  a  bankrupt  merchant  are  $18,900, 
and  his  assets  are  $8,344.  How  much  will  a  creditor  receive 
on  a  debt  of  $2,250  ? 

19.  How  much  will  .87^  of  a  barrel  of  apples  cost,  at  $1 
for  f  of  a  barrel  ? 

W.  If  5|  cd.  of  limestone  will  produce  450.56  bu.  of  lime, 
how  many  cords  of  stone  will  be  required  to  produce  l,663f 
bu.  of  lime  ? 

21.  Bought  16  pieces  of  merino,  each  piece  containing  42^ 
yd.,  at  the  rate  of  $36.45  for  27  yd.     What  did  it  cost  ? 

22.  I  borrowed  of  a  friend  $675  for  8  months  ;  afterward 
I  lent  him  $450.  How  long  must  he  keep  the  loan,  to  balance 
the  favor  ? 

23.  If  8i  tons  of  coal  cost  $46.75,  what  will  4  tons  cost  ? 
2Jf.  At  $5  for  I  yd.  of  cloth,  what  is  the  value  of  16^  yd.  ? 

25.  If  a  bird  can  fly  9|  miles  in  \  of  an  hour,  how  far  can 
it  fly  in  9^  hours,  at  the  same  rate  ? 

26.  If  8  men  mow  a  meadow  in  5  days,  in  how  many  days 
will  3  men  mow  it  ? 

27.  If  300  sheep  require  150  A.  156  sq.  rd.  of  pasture,  how 
many  acres  will  850  sheep  require  ? 

28.  If  156  pocket-knives  cost  $168,  how  much  will  57  cost  ? 


RATIO  AND  PROPORTION.  339 

III.  Compound  Peopoetion, 
16S,  Compound  I^roportion  is  an  equality  of  a  com- 
pound and  a  simple  ratio,  or  of  two  compound  ratios. 
4:8)  (4:8)        (6:12)  are  compound 

^2  .  3  p:  10  :  5,  and  I  j2  :  3  i  ••  (  8  :  2  f  proportions. 
10 6*  A  compound  ratio  is  reduced  to  a  simple  one,  by- 
multiplying  all  the  antecedents  together  for  a  new  antece- 
dent, and  all  the  consequents  together  for  a  new  consequent. 
(See  155,)  Hence,  the  principle  {160)  applies  equally  to 
compound  proportion. 

Peoblems. 
Find  the  unknown  term  in  each  of  the  following  propor- 
tions : 

4:  12 


}::3,- 


•      9:18 

12:3 

2.     3:4    }-  ::54; 

4:5 

-$10:  $.06,       ^^ 

^'    dbu?    &Q     >•  : :  1 0  oz. :  —  oz. 


}::: 


~:21    ^ 
6,   48:1.6  >  ::8i:2iio. 
1.5:. 2     ) 

12:3      )    ^.  j28:  — 
•     4:25    f  ••(  20:5. 


167*  Ex.  If  6  weavers  weave  81  yards  of  cassimere  in  9 
hours,  how  many  yards  will  10  weavers  weave  in  8  hours  ? 

Explanation.  —  Process. 

quired   term    is     ^  y^eavers  :  1 0  weavers  )    ..^^      ,  ._^ 
yards,  I  write  81         9  hours :    8  hours      )    "         ^  "       ^ 
yards  for  the  first  term  S 

of  the  second  ratio,  or      ^  ^ 

the  third  term  of  the      ^  r(       a       a  -^      i 
proportion.  ^Ij^lU^lljlil  =  U  0  yd. 

The  required  number  of  J6f  x  J9^ 

yards  depends  upon  the  % 

number    of    weavers, 

and  the  number  of  hours.  I  therefore  arrange  the  other  given 
numbers  in  pairs,  as  the  terms  of  the  compound  ratio.  Thus,  6 
weavers  weave  81  yards,  10  weavers  will  weave  more;  hence, 
6  weavers  :  10  weavers;  and  in  9  hours  they  weave  81  yards,  in 
8  hours  they  will  weave  less ;  hence  9  hours  :  8  hours. 

Having  made  the  statement,  I  solve  the  proportion,  and  obtain  120 
yards,  the  required  term. 


340  SUPPLEMENT. 

168.  Rule  for  Compound  Proportion. 
I.  For  the  third  term  :  —  Write  the  number  that  is  of  the 
same  kifid  as  the  required  term, 

11.  For  the  compound  ratio  :  —  Write  each  two  of  the  given 
numbers  that  are  of  the  same  kind,  as  a  couplet,  writing  the 
greater  of  the  two  for  the  second  term,  when  t/te  required  term 
is  to  be  greater  than  the  third  term  ;  and  the  less  of  the  two 
for  the  second  term,  when  the  required  term  is  to  be  less  than 
the  third  term, 

III.  To  find  the  fourth  or  required  term  : — 
Multiply  all  the  second  and  third  terms  together^  and  divide 
the  product  hy  the  product  of  the  first  terms. 

Adapt  remarks  a,  b,  after  rule  for  simple  proportion,  104:^  to  ap- 
ply to  compound  proportion. 

Note.— In  solving  problems,  many  teachers  write  ^  ^  V^*    '^  V^'  ^  V^' 

the  second  and  third  terms  on  the  right  of  a      %   fHi 


vertical  line,  and  the  first  terms  on  the  left —  ry 

as  here  shown— in  preference  to  writing  them 

above  and  below  a  horizontal  line.  / 


U5 

8 
120  yd. 


Problems. 

1.  If  the  floor  of  a  room  13  ft.  6  in.  by  18  ft.  cost  $12.16, 
what  is  the  cost  of  the  floor  of  a  room  15  ft.  9  in.  by  16  ft. 
6  in.? 

2.  If  a  cistern-pump  factory,  running  10  hours  per  day, 
makes  3,060  pumps  in  27  days,  how  many  pumps  will  the 
same  factory  make  in  63  days,  running  12|^  hours  per 
day? 

3.  I  paid  $675  for  a  building  lot  3f  rd.  front  by  11  rd.  deep. 
Afterward  I  bought  another  lot  3^  rd.  front  by  13|^  rd.  deep, 
at  the  same  rate.     How  much  did  I  pay  for  the  second  lot  ? 

4.  A  pile  of  gypsum  stone,  1,225  ft.  long,  12  ft.  wide,  and 
4  ft.  high,  was  drawn  to  a  plaster  mill  by  10  teams  in  21  days. 
At  the  same  rate,  how  many  days  will  it  take  8  teams  to 
draw  a  pile  1,607  ft.  long,  16  ft.  wide,  and  5  ft.  high? 


RATIO    AND    PROPORTION.  341 

5.  If  $93.87  is  the  interest  of  $360  for  3  yr.  8  mo.  21  da., 
$13.02  is  the  interest  of  what  principal  for  1  yr.  6  mo. 
18  da.? 

6.  In  building  a  tight  board  fence  1  mi.  long  and  11  ft. 
high,  around  a  fair  ground,  480  8 -inch  boards  were  used. 
How  many  10-inch  boards  will  be  required  to  build  a  fence 
2,730  yd.  long  and  8  ft.  high? 

7.  If  5  men  make  150  pairs  of  boots  in  6  days,  how  many 
pairs  will  3  men  make  in  4  days  ? 

8.  An  investment  of  $800  gains  $70  in  15  months.  At 
the  same  rate,  how  much  will  an  investment  of  $356  gain  in 
8  months  ? 

9.  If  2,100  ft.  of  ij-inch  flooring  is  required  for  the  floors 
of  a  certain  house,  how  much  1^-inch  flooring  will  be  re- 
quired for  the  floors  of  another  house  2^  times  as  long  and 
1^  times  as  wide  ? 

10.  Four  railroad  passengers  rode  36  mi.,  60  mi,,  72  mi., 
and  96  mi.  respectively.  They  paid  in  proportion  to  the 
distances  they  rode,  and  the  sum  of  their  fares  was  $9.24. 
How  much  did  each  man  pay  ? 

11.  The  capacity  of  a  bin  24  ft.  long,  4  ft.  6'  wide,  and  4  ft. 
8'  deep  is  405  bushels.  What  is  the  capacity  of  a  bin  10^ 
ft.  long,  6  ft.  w^ide,  and  5  ft.  deep  ? 

12.  If  3  men  dig  453  bushels  of  potatoes  in  a  week,  how 
many  bushels  can  2  men  dig  in  5  days  ? 

13.  If  three  men  can  lay  a  sidewalk  240  ft.  long  and  6 
ft;  wide,  in  15  days,  in  how  many  days  can  5  men  lay  a 
walk  180  ft.  long  and  4  ft.  wide  ? 

H.  A  pile  of  4-foot  wood  244  feet  long  and  5  feet  high, 
was  sold  for  $152.50.     What  was  the  price  per  cord  ? 

(8  ft.  by  4  ft.  by  4  ft.  =lcd.) 

15.  If  in  16  days  of  9  hours  each,  9  bricklayers  lay  a  wall 
96  ft.  long,  21  ft.  high,  and  1|-  ft.  thick,  in  how  many  days 
of  Hi  hours  each  can  12  bricklayers  lay  a  wall  126  ft.  long, 
28  ft.  high,  and  l^  ft.  thick  ? 


[342] 


§  19.   POWERS  AND  ROOTS. 
I.  Involution. 
169*  A  power  is  the-  product  of  equal  factors. 

a.  The  second  power  or  square  is  the  product  of  two  equal 
factors. 

b.  The  third  power  or  cube  is  the  product  of  three  equal 
factors. 

c.  The  second  power  or  square  of  4  is  16  (=4x4); 
and  the  third  power  or  cube  of  4  is  64  (=  4  x  4  x  4). 

170*  Involution  is  the  process  of  finding  powers. 

17 1»  The  sign  of  involution  is  an  index  or  exponent , 

which  is  a  small  figure  placed  at  the  right  of,  and  above,  a 
number. 

An  exponent  shows  the  required  power  of  a  number.    Thus, 
2P  signifies  the  square  of  21 ;  3.5*  signifies  the  cube  of  3.5. 
172.  Complete  and  learn  the  following  table  : 


Numbers. 

Squares. 

Cubea 

1 

S 

S 

4 

5 

6 

7 

8 

9 

Numbers. 

Squares. 

Cubea 

.1 

.2 

.3 

.k 

.5 

.6 

.7 

.8 

.9 

Problems. 


Find   the   following  (J  ^J,' 

indicated  powers:    j    *     \ 
^  [S.  12^ 


7. 

iW 

10,       .17^ 

8. 

(1)^ 

11,  20.16^ 

9. 

(15§)^ 

12,  (l.03y 

i.  901' 

5.  218' 

6.  139' 

II.  Evolution. 
17 3 •  A  root  of  a  number  is  one  of  the  equal  factors  that 
produce  the  number. 

a.  The  square  root  of  a  number  is  one  of  the  two  equal  fac- 
tors that  produce  the  number. 

b.  The  cuhe  root  of  a  number  is  one  of  the  three  equal  factors 
that  produce  the  number. 

C.  The  square  root  of  64  is  8  (64  =  8  x  8) ; 
and  the  cube  root  of  64  is  4  (64  =  4  x  4  x  4). 


POWERS   AND    R0  0T8.  343 

174,  Evolution  is  tlie  process  of  finding  roots. 
17 d*  The  sign  of  evolution  is  -y/;  it  is  called  the  radical 
sign, 

a.  ^/  placed  before  a  number,  indicates  that  tlie  square  root  of  the 
number  is  to  be  found. 

b.  -^  placed  before  a  number,  indicates  that  the  cube  root  of  the 
number  is  to  be  found. 

C.  V^5  indicates  that  the  square  root  of  25  is  to  be  found  ;  and 
-^64  indicates  that  the  cube  root  of  64  is  to  be  found. 

d#  A  perfect  power  is  a  number  whose  exact  root  can  be  found ; 
an  imperfect  power  is  a  number  whose  exact  root  can  not 
be  found. 

III.  Extractio:n^  op  the  Square  Root. 

176*  Extraction  of  the  square  root  is  the  process  of 

finding  one  of  the  two  equal  factors  that  produce  a  number. 

The  least  and  the  greatest  integer  that  can  be  expressed  by  one 

figure  are,  respectively,  1  and  9 ;  and  by  two  figures  10  and  99. 

The  least  and  the  greatest  decimal  that  can  be  expressed  by  one 

figure  are,  respectively,  .1  and  .9;  and  by  two  figures,  .01  and  .99. 

The  squares  of  tliese  numbers  are 

1'=:      1  10'  rr:       100  .l'  =r:  .01  .01'  =  .0001 

9^=81  99' =9,801  .9' =  .81  .99' =  .9801 

Comparing  these  numbers  with  their  squares,  we  see  that 

1.  The  square  of  an  integer 

j   one  figure  )  is  expressed  j  one  or  two  figures, 
expressed  by  -j  ^^^^  fig^^res  j  by  ( three  or  four  figures. 

2.  The  square  of  a  decimal 

,  ,      C    one  fi2:ure   )  is  expressed  ( two  figures, 
expressed  by  •{  ^       ,,  ^         J-  f  ■<  .        J^ 

^  "^   (  two  ngures  )  by  ( four  figures. 

and  so  on,  of  greater  numbers.     Hence, 

17  7 »  Principle  I.   The  square  of  any  integer  is  expressed 

hy  twice  as  many  figures  as  the  integer,  or  one  less  than  twice 

as  many ;  and  the  square  of  any  decimal  is  expressed  hy 

ticice  as  many  decimal  figures  as  the  decimal. 

From  this  principle  it  is  evident  that  tlie  number  of  two-figure 

periods  in  a  number  equals  the  number  of  places  in  its  square  root. 


344  SUPPLEMENT. 

Squaring  the  integers  2,  20,  and  25;  and  9,  90,  and  99;  and  the 
decimals  .9,  .25,  and  .99,  we  have 

2'=      4  9'=       81  .9'    =.81 

20'  =  400  90'  =  8,100  .25'  =  .0625 

25'  =  625  99*^  =  9,801  .99'  =  .9801 

Separating  the  squares  into  two-figure  periods,  and  comparing  the 
numbers  with  their  squares,  we  see  that 

1.  The  square  of  the  left-hand  digit  is  wholly  in  the  left-hand 
period.     And 

2,  The  square  of  the  left-hand  digit  is  the  greatest  square  in 
the  left-hand  period.     Hence, 

178.  Principle  II.  The  square  of  the  left-hand  order  of 
units  of  a  number  is  wholly  in  the  left-hand  period  of  the 
power,  and  is  the  greatest  square  i?i  that  period, 

179.  To  determine  the  parts  that  make  up  the  square  of 
a  number,  we  will  square  56  by  different  processes. 

56=^50-f  6;  and  56'  =  56x56,  or  50 -f  6  multiplied  by  50  +  6. 

PIRST  PROCBSS.  8BC0ND  PROCESS.  THIRD  PROCESS. 

56=  50  +  6=:  50-{-6 

56=.  50-^6=  50-\-6 


300 -{•86=  (50x6) +  6' 

280    =2,500  +  800  =50^+         {50x6) 


8, 186  =  2, 500 +  600 +  86  =  50' -\- 2  X  (50x6) +  6' 
In  the  first  process  56  is  squared  by  the  common  method  of  mul- 
tiplication. 
In  the  second  process  the  products  of  the  units  of  the  different 

orders  are  written  separately. 
In  the  third  process  the  factors  that  make  up  the  different  parts  of 
the  product  are  kept  separate,  the  multiplications  and  additions 
being  indicated  only.     This  third  process  fully  illustrates 

180,  Principle  III.  The  square  of  a  number  regarded 
as  tens  and  ones  equals  the  square  of  the  tens,  plus  two  times 
the  product  of  the  tens  and  ones,  plus  the  square  of  the  ones. 

The  several  steps,  in  their  order,  in  extraction  of  the  square  root;, 
are  based  on  the  three  principles  now  given. 


POWERS   AND    ROOTS. 


345 


181.  Ex.  1.  Extract  the  square  root  of  3,136. 


Explanation. —  Separating 
the  number  into  periods 
of  two  figures  each,  I  find 
that  the  root  will  be  ex- 
pressed by  two  figures. 

I  write  25,  the  greatest  square 
in  the  left-hand  period,  un- 


Process. 


2x50  =  100 
2x50-^6  =  106 


S1'S6\56 
25 

6S6 

636 


der  the  period;  and  5, the  square  root  of  25,for  the  tens  of  the  root. 

Subtracting  25  from  31, 'and  to  the  remainder  annexing  the  next 
period,  I  have  636.  This  number  is  made  up  of  two  times  the 
product  of  the  tens  and  the  ones  of  the  root,  plus  the  square  of 
the  ones ;  i.  e.,ot  2  x 50 x ones + ones ^. 

Dividing  636  by  100  (=  2  x  50)  regarded  as  a  trial  divisor,  I  obtain 
6,  which  I  write  for  the  ones  of  the  root ;  and  since  100  =  2x5 
tens,  and  636  =  2x5  tens  x  the  ones,  +  the  square  of  the  ones,  I 
also  add  6  to  100,  the  trial  divisor,  making  106,  the  complete  di- 
visor. Multiplying  this  complete  divisor  by  6,  the  ones  of  the 
root,  I  have,  1st, — 6x6,  or  the  square  of  the  ones;  and  2d, — 
6  x  100,  or  two  times  the  product  of  the  tens  and  the  ones.  The 
product,  636,  is  the  same  as  the  dividend;  and  56  is  the  square 
root  required. 
In  extraction  of  the  square  root,  only  two  periods  of 
figures  are  considered  in  connection.     Hence, 

In  obtaining  any  figure  of  the  root  except  the  first,  the  figure  or 
figures  of  the  root  already  found  are  regarded  as  tens,  and  the 
figure  sought  as  ones ;  ^.  e. ,  each  succeeding  figure  of  the  root  is 
found  in  the  same  manner  as  the  second  figure  of  a  root  ex- 
pressed by  two  figures.  (See  the  solution  of  Ex.  2.) 
Ex.  2.  Extract  the  square  root  of  56,192.'7025. 


Full  Process. 

5' 61' 92,70' 25  \237. 05 

± 
2x20=Jf0    161 
2x20-^S=ks\l29 

2x2S0=Jf60   8292 
2x230+7=^67  ^269 

iSX    2370=  U7U0 
\^X  2-3700=  U7 WO 
^x2S700+5=47m  

P2 


237025 

^37025 


Common  Process. 


5' 61' 92,70' 25 

A 
161 
129 

3292 
3269 


237025 

237025 


237.05 


^ 


J^67 


i7Jf05 


346  SUPPLEMENT. 


Ex.  3.  Extract  the  square  1289  _  V^89  _17 

root  of  Iff.  ^  ^^^^"'-  V  6¥5 '-^/^5^J5 

182*  Rule  for  Extraction  of  Square  Root. 

I.  To  determine  the  number  of  figures  in  the  root : — 
Separate  the  number  into  periods  of  two  figures  each,  begin- 
ning  at  07ies  in  an  integer^  and  counting  from  ones  in  a 
decimaL 

II.  To  find  the  first  figure  of  the  root: — 

1.  Find  the  root  of  the  greatest  square  in  the  left-hand 
period,  and  write  it  for  the  first  figure  of  the  root. 

2.  Subtract  this  square  from  the  first  period;  and  to  the 
remainder  annex  the  next  period  for  a  dividend, 

III.  To  find  the  second  figure  of  the  root: — 

1.  Double  the  root  already  found,  considered  as  tens,  for 
a  trial  divisor,  by  which  divide  the  dividend ;  icrite  the  re- 
sult for  the  second  figure  of  the  root,  and  also  iri  the  place  of 
ones  in  the  trial  divisor,  thus  forming  the  complete  divisor, 

2.  Multiply  the  complete  divisor  by  the  secoiid  figure  of 
the  root ;  subtract  the  product  from  the  dividend ;  and  to  the 
remainder  annex  the  next  period  for  a  dividend, 

IV.  To  find  the  succeeding  figures  of  the  root: — 
Proceed  with  tJie  second,  and  with  each  succeeding  divi- 
dend, in  the  same  manner  as  with  the  first. 

a.  If  any  dividend  is  less  than  the  divisor: — Wnte  a  cipher  in  the 
root,  aiid  also  on  the,  right  of  the  diciwr;  and  annex  the  next  period 
to  the  dividend,  for  a  new  dividend. 

b.  If  there  is  a  remainder  after  all  the  periods  have  been  used : — 
Annex  periods  of  decimal  ciphers,  and  extend  the  work  to  any  re- 
quired degree  of  exactness. 

C.  When  the  given  number  is  an  imperfect  power: — Wnte  +  after 
tlie  root,  to  indicate  that  the  root  is  not  exact. 

d.  If  the  right-hand  decimal  period  contains  hut  one  figure: — 
Annex  a  decimal  cipher. 

e.  To  extract  the  square  root  of  a  mixed  fractional  number  :-^JFVrsf 
reduce  it  to  a  mixed  decimal  number,  or  to  an  improper  fraction. 


POWERS   AND    ROOTS.  347 

Problems. 
Extract  the  square  root  of        |      Find  the  value  of 


10.  Vfif . 

11.  ^231114. 

12.  Vs.  I  13.  V'J'S. 


1.  2,916.  4.  .0784.  7.   '/56.25. 

2.  782,169.  5.  .186624.      <?.   ^3, 685. 582681. 

3.  29,735,209.      6.  f|f.  5.  V. 000289. 

H.  What  is  the  length  of  one  side  of  a  square  piece  of 
land  that  contains  9,312.25  sq.  ft.  ? 

15.  The  surface  of  the  water  in  a  square  reservoir  meas- 
ures 12,488-3ig-  sq.  yd.    How  long  is  one  side  of  the  reservoir? 

16.  What  are  the  dimensions  of  a  box,  the  entire  area  of 
its  6  equal  square  sides  being  522|-  cu.  in.? 

17.  A  park  is  f  as  wide  as  it  is  long,  and  its  area  is  32,256 
sq.  ft.     What  are  its  dimensions  ? 

18.  What  is  the  length  of  one  side  of  a  square  field  whose 
area  is  21rjig-  acres  ? 

19.  An  oblong  field  containing  12.1  acres,  is  4  times  as 
long  as  it  is  wide.     What  are  its  dimensions  ? 

IV.   EXTEACTION-   OF   CuBE   RoOT. 

183.  Extraction  of  the  cube  root  is  the  process  of  find- 
ing one  of  the  three  equal  factors  that  produce  a  number. 

The  cubes  of  1  and  9, 10  and  99,  .1  and  .9,  .01  and  .99,  respective- 
ly, are 

l^=       1        10'=      1,000        .I'rir.OOl        .01' =  .000001 
9' =  729        99^  =  970,299        .9' =  .729        .99' =  .970299 
Comparing  these  numbers  and  their  cubes,  we  see  that 

1.  The  cube  of  an  integer 

expressed  j  one  figure  )  is  expressed  j  one,  two,  or  three  figures, 
by        ( two  figures  )  by  (  four,  five,  or  six  figures. 

2.  The  cube  of  a  decimal 

,  ,     j  one  figure  )  is  expressed  j  three  decimal  figures. 
^  ■       ( two  figures  )  by  (  six  decimal  figures. 

And  so  on,  of  greater  numbers.     Hence, 


348  SUPPLEMENT. 

IS 4*  Principle  I.  The  cube  of  any  iyiteger  is  expressed 
by  three  times  as  mariy  figures  as  the  integer,  or  one  or  two 
less  than  three  times  as  many  y  and  the  cube  of  any  decimal 
is  expressed  by  three  times  as  many  decimal  figures  as  the 
decimal. 

From  this  principle  it  is  evident  that 
The  number  of  three-figure  periods  in  a  number  equals  the  num- 
ber of  places  in  its  cube  root. 

Cubing  the  integers  2,  20,  and  25;  9,  90,  and  99;  and  the  decimals 
.9,  .25,  and  .99,  we  have 

2'  =  8  9^  =  729  .9^  =  .729 

20=^==    8,000  90^=729,000  .25' =  .015625 

25'  =  15,625  99^  =  970,299  .99'  =  .970299 

Separating  the  cubes  into  three-figure  periods,  and  comparing  the 
numbers  with  their  cubes,  we  see  that 

1,  The  cube  of  the  left-hand  digit  is  wholly  in  the  left-hand 
period.     And 

2.  The  cube  of  the  left-hand  digit  is  the  greatest  cube  in  the 
left-hand  period.     Hence, 

185.  Principle  II.  The  cube  of  the  left-hand  order  of 
units  of  a  number  is  wholly  in  the  left-hand  period  of  the 
poioer,  and  is  the  greatest  cube  in  that  period, 

186.  To  determine  the  parts  that  make  up  the  cube  of  a 
mimber,  we  will  cube  35  by  different  processes. 

35  =  30  +  5;  and  35^  =  35x35x35,01  the  product  of  30  +  5, 
30  +  5,  and  30  +  5. 

First  Process.      Second  Process.  Third  Process. 

SS=                                   30+     5=  80+5 

35:=                                  30+     6=  30+5 

175=                                150+  25=  {30^5)+5'' 

105  =                      900  +  150          =  30''+       {30X5) 

1285=                       900+300+  25=  30''+2x{30x5)+5^ 

35=  30+     5=  30+5 

6125=                   ^00  +  1500  +  125=  {30^  x5)+2x  {30  X  5^)+5* 

3675  =   27000+  9000+  750          =  30^+2  X  {30""  x5)+       {30  X  5^) 

Jt2S75=  27000+13500+2250+125=  30^+2x{30''x5)+3x{30xS^)+6* 


POWERS  AND    ROOTS,  349 

The  several  parts  of  the  cube,  reading  from  the  left,  are 
1st.  The  cube  of  the  tens,  27,000 

2d.  Three  times  the  square  of  the  tens  X  the  ones,  13,500 

3d.  Three  times  the  tens  X  the  square  of  the  ones,  2,250 

/i,th.  The  cube  of  the  ones,  125 

That  is,  35^  =  42,875 
187*  Principle  III.  The  cube  of  a  number,  regarded  as 
tens  and  ones,  is  equal  to  the  cube  of  the  tens,  plus  three  times 
the  product  of  the  square  of  the  tens  and  the  ones,  plus  three 
times  the  product  of  the  tens  and  the  square  of  the  ones,  plus 
the  cube  of  the  ones. 

188.  The  several  steps,  in  their  order,  in  extraction  of 
the  cube  root,  are  based  on  the  three  principles  now  given. 

189.  Ex.  1.     Extract 
the  cube  root  of  42,875.  Jf.2'87 5\35 

Explanation.  —  Separat- 
ing the  number  into  pe- 
riods of  three  figures 
each,  I  find  that  the 
root  will  be  expressed 
by  two  figures. 

I  write  27,  the  greatest 
cube  in  the  left-hand 

period,  under  the  period ;  and  3,  the  cube  root  of  27,  for  the 
tens  of  the  root. 

Subtracting  27  from  42,  and  to  the  remainder  annexing  the  next 
period,  I  have  15,875.  This  number  is  made  up  of  three  times 
the  product  of  the  square  of  the  tens  and  the  ones,  plus  three 
times  the  product  of  the  tens  and  the  square  of  the  ones,  plus  the 
cube  of  the  ones;  i.  e.,  of  3 x 30*^ x ones,  +3 x 30 x ones^  +ones^ 

Considering  the  first  figure  of  the  root  as  tens,  and  multiplying 
its  square  by  3, 1  have  2,700  for  a  trial  divisor.  Dividing  15,875 
by  this  tr.ial  divisor,  I  have  5  for  the  ones  of  the  root. 

Adding  to  the  trial  divisor  450  (=3  x  30  x  5)  and  also  25  (=  5=),  I 
have  3,175,  the  complete  divisor. 

Then,  multiplying  this  complete  divisor  by  5,  the  ones  of  the  root, 
I  have  15,875,  which  is  made  up  of,  Ist,—^  x  5  x  5,  or  ones'; 
^d,—^ x 30 X 5 X 5,  or  3 x tens x ones^ ;  and,  3d,—^x 30 x 30 x 5, 
or  3  X  tens'*  x  ones. 

I  have  now  used  all  of  the  given  number,  and  35  is  the  cube  root 
required. 


Process. 

i2'875\ 

27 

S  x30''=:2,700 

15875 

3xS0xS=z    450 

5'=       25 

3,175 

15875 

350 


SUPPLEMENT. 


In  extraction  of  the  cube  root,  only  two  periods  of  figures  are 

considered  in  connection.     Hence, 
Any  figure  of  the  root  after  the  second,  is  found  in  the  same  manner 

as  tJie  second  figure  of  a  root  expressed  hy  two  figures. 
This  is  fully  shown  in  the  solution  of  Ex.  2. 
Ex.  2.    What  is  the  cube  root  of  9,938,375? 


Full  Process. 


9'938'376\215 
8 


Trial  divisor,  3x20'^  =  1,200  1938 
3x20x1=      60 

P= i 

Complete  divisor,  1,261 1261 

Trial  divisor,  3  X  210^=132,300\677375 
3x210x5=     3,150 

5^= 25 

Complete  divisor,  135,4.75  677375 


Common  Process. 


9'938'375 

8 

1,938 

215 
1,261 

1,261 

677,375 

135,475 

677,375 


Process. 


\^1W608 


imo8 

1331 


--  —  AS 
^1331  11  ~'^^^' 


Ex.  3.  Find  the  cube  root  of  105^^ 

190.  Rule  foe  Extraction  of  Cube  Root. 
I.  To  determine  the  number  of  figures  in  the  root : — 

Separate  the  number  into  three-figure  periods,  heginning 
AT  ones  in  an  integer,  and  counting  from  ones  in  a  decimal, 
11.  To  find  the  first  fig-ure  of  the  root : — 

1.  Find  the  root   of  the  greatest  cube  in  the  left-hand 
period,  and  write  it  for  the  first  figure  of  the  root, 

2,  Subtract  this  cube  from  the  first  period,  and  to  the  re- 
mai7ider  annex  the  next  period  for  a  dividend. 

III.  To  find  the  second  figure  of  the  root : — 
1,  Square  that  part  of  the  root  found,  considered  as  tens, 
and  midtiply  the  square  by  3,  for  a  trial  divisor,  by  which 
divide  the  dividend;  and  write  the  result  for  the  second  fig- 
ure of  the  root. 


POWERS  AND   ROOTS.  351 

2.  To  the  trial  divisor  add  S  times  the  product  of  the  tens 
and  ones  of  the  root  already  found,  and  also  the  square  of  the 
ones,  for  a  complete  divisor. 

S.  Midtipfly  the  complete  divisor  by  the  second  figure  of 
the  root ;  subtract  the  product  from  the  dividend ;  and  to  the 
remainder  annex  the  next  period,  for  a  second  dividend. 

IV.  To  find  the  succeeding  figures  of  the  root : — 

Proceed  with  the  second,  and  each  succeeding  dividend,  in 
the  same  manner  as  with  the  first,  until  all  the  periods  are  used, 

a.  If  any  dividend  is  less  than  the  divisor  :  —  Write  a  ciplier  in  the 
root;  annex  two  ciphers  to  the  trial  divisor,  for  a  neio  divisor;  and 
the  next  period  to  the  dividend,  for  a  new  dividend. 

b.  Adapt  remarks  a,  h,  c,  d,  e,  after  rule  for  extraction  of  square 
root,  Art.  182,  to  apply  to  extraction  of  cube  root. 

Problems. 
Extract  the  cube  root  of  Find  the  value  of 

1.  148,877.      4.  .658503.      7.    -^178.453547.       10.    ^Z-f^. 

2.  39,304.      5.  .000512.      8.    -^128.024064.       11.   -^10^1^. 
S.  .006859.      6.  300.763.      9.    -^109,095.488.      12.   -^5^. 

18.  The  capacity  of  a  cubical  cistern  is  6  cu.  yd.  19  cu.  ft. 
1,664  cu.  in.     What  is  its  size  ? 

IJ^.  Find  one  of  the  three  equal  factors  of  118,805,247,296. 

15.  The  contents  of  a  wall  are  216  cu.  ft.,  its  height  is  2 
times  its  thickness,  and  its  length  is  48  times  its  height. 
What  are  its  dimensions  ? 

What  is  the  inside  measurement 

16.  Of  a  cubic  box  that  will  hold  a  bushel  of  wheat  ? 
11.  Of  a  cubic  can  that  will  hold  a  gallon  of  oil  ? 

18.  What  is  one  dimension  of  a  cubical  bin  that  will  hold 
the  same  number  of  bushels  of  wheat  as  a  bin  that  is  12  ft. 
long,  7  ft.  wide,  and  4  ft.  deep  ? 

19.  Find  the  inside  measurement  of  a  cubic  box  whose 
capacity  equals  that  of  a  box  6  ft.  6  in.  long,  5  ft.  5  in.  wide, 
and  4  ft.  4  in.  deep. 


[352] 

§  20.  MEASUREMENTS. 
I.  Applications  of  Square  and  Cube  Root. 
101*  The  most  important  applications  of  square  root 
and  cube  root  are,  in  finding  the  sides  of  right-angled  trian- 
gles, and  in  computations  of  like  parts  of  similar  figures  and 
similar  volumes. 

The  principles  upon  which  the  applications  of  these  proc- 
esses are  based  are  demonstrated  in  geometry. 
To  find  the  hypothenuse  of  a  triangle. 

192*  The  hypothenuse  of   a   right-angled 
triangle  is  its  longest  side. 

103.  The  hase  and  perpendicular  of  a  right- 
angled  triangle  are  the  two  sides  that  include  the  right  angle. 

a.  In  the  right-angled  triangle  ahc,  the  side 
a  c  is  the  hypothenuse,  the  side  ab  m  the 
base,  and  the  side  c  6  is  the  perpendicular. 

b.  The  two  shorter  sides  are  also  called  the 
legs  of  the  triangle. 

The  area  of  the  square  described  (or  drawn) 
on  the  hypothenuse  of  a  right-angled  triangle, 
is  equal  to  the  sum  of  the  areas  of  the  squares 
described  on  the  other  two  sides.     Hence,  Figure  n. 

194,  The  hypothenuse  of  a  right-angled  triangle  equals 
the  square  root  of  the  sum  of  the  squares  of  the  other  two  sides. 

Problems. 

1.  My  garden  is  a  rectangle  1 16  ft.  2  in.  long,  and  87  ft.  9  in. 
wide.    What  is  the  distance  between  the  diagonal  corners  ? 

^.  Find  the  length  of  a  hand  rail  to  a  straight  flight  of 
19  stairs,  each  10-J^  in.  wide  and  Vi  in.  high. 

S,  The  four-sided  roof  of  a  house  is  24  ft.  by  30  ft.  at  the 
eaves,  and  the  peak  is  9  ft.  higher  than  the  eaves.  What  is 
the  distance  from  either  corner  of  the  roof  to  the  peak  ? 

4.  A  ladder  39  ft.  long  reaches  the  top  of  a  building  36  ft. 
high.     How  far  from  the  building  does  its  foot  stand  ? 


ME  A  S  U  RE  MEN  TS. 


353 


5,  What  is  the  distance  from  the  top  of  a  building  72  ft. 
high,  to  the  opposite  side  of  a  street  82  ft.  wide  ? 

6,  The  distance  between  the  diagonal  corners  of  two  in- 
tersecting streets  of  equal  width,  is  88  ft.  How  wide  are 
the  streets  ? 

7,  Find  the  side  of  a  cube  whose  diagonal  is  43.3  inches. 

8,  The  rafters  are  16.5  ft.  long,  and  the  gable  is  26.4  ft. 
wide.     How  high  above  the  eaves  is  the  gable  peak  ? 

9,  Three  iron  rods  extend  from  the  top  of  a  derrick  30  ft. 
high,  to  the  ground  at  the  distances  of  40  ft.,  72  ft.,  and  133 
ft.  4  in.  from  the  foot  of  the  derrick.  Find  the  lengths  of 
these  iron  rods. 

10,  What  are  the  dimensions  of  the  ends  of  the  largest 
stick  of  square  timber  that  can  be  cut  from  a  log  22  inches 
in  diameter  ? 

II.  Peisms,  Cylinders,  Pyramids,  Cones,  and  Spheres. 
195*  A  polygon  is  a  plane  figure 
whose  sides  are  straight  lines. 

196*  A  prison  is  a  solid  whose  bases 
are  equal  parallel  polygons,  and  whose 
sides  are  parallelograms. 

The  name  of  a  prism  is  determined  from 
the  form  of  its  bases  or  ends. 
197.  A  cylinder  is  a  round  solid  of  uniform  diameter, 
whose  bases  are  equal  parallel  circles. 

a«  The  axis  of  a  prism  or  of  a  cylinder  is  a  straight  line  that 

joins  the  centres  of  its  bases. 
b.  A  right  prism  or  a  right  cylinder 
is  one  whose  axis  is  perpendicular  to  its 


Figure  18.        Figure  19. 


19S»  A  pyramid  is  a  solid  whose 
base  is  a  polygon,  and  whose  sides  are 
triangles  terminating  in  a  point  or  vertex. 

199*  A  cone  is  a  round  solid,  whose 
base  is  a  circle,  and  whose  top  terminates  in  a  point  or  vertex. 


Figure  20.      Figure  21. 


354:  SUPPLEMENT. 

a«  The  axis  of  a  pyramid  or  of  a  cone  is  a  straight  line  that 

joins  the  vertex  with  the  centre  of  its  base. 
b.  A  right  2>y^€itnid  or  a  right  cone  is  one  whose  axis 

is  perpendicular  to  its  base. 
C.  The  altitude  of  a  pyramid  or  a  cone  is  the  perpendicular 

height  of  it»  vertex  above  its  base. 

d.  The  slant  height  of  a  pyramid  is  the  distance  from  the  ver- 
tex to  the  middle  of  one  side  of  its  base. 

e.  The  slant  height  of  a  cone  is  the  distance  from  its  vertex 
to  the  circumference  of  its  base. 

Case  I.  To  find  the  area  of  a  prism,  cylinder,  pyra- 
mid, or  cone, 

200*  The  entire  surface  of  a  prism  consists  of  its  bases 
and  its  sides.    And 

The  entire  surface  of  a  cylinder  consists  of  its  bases  and  its 
curved  surface  (which  is  equal  to  a  parallelogram  whose  two 
dimensions  are  the  length  and  circumference  of  the  cylinder). 

Ex.  1.  What  is  the  area  of  a  prism  8  feet  long  and  14 
inches  square  ? 

Full  Solution. 
2  X  IJfX  IJfSq.  in,  —      39  2  sq.  in.,  area  of  the  bases. 
J^X  Hx  9  6  sq.in.  —  5,3  7  6  sq.  in.,  area  of  the  four  sides. 

Total  area,  5,7  68  sq.  in.  =  Jf.  sq.  yd.  ^  sq.ft.  8  sq.  in. 

Ex.  2.  What  is  the  area  of  a  cylinder  8  feet  long  and 
14  inches  in  diameter? 

Full  Solution. 
8^  X  lJ^in.:=  JfJf  in.,  circumference. 
2  X  -^^^  X  ^^  sq.  in.  =       3  08  sq.  in.,  area  of  the  bases. 
^^  X  9  6  sq.in.  =  4-1^  ^  -i-  sq.  in.,  area  of  curved  surface. 
Total  area,  Jf,5  3  2  sq.  in.=3sq.  yd.  Jf  sq.ft.  68  sq.  in. 

The  number  of  units  in  the  entire  area  of  a  2)risni  or  a 
cylinder  is  equal  to  the  sum  of  the  units  in  its  bases,  plus 
the  U7iits  in  its  lateral  surface. 


MEASUREMENTS.  355 

20 !•  The  entire  surface  of  a  pyramid  consists  of  its  base 
and  its  lateral  surfaces.     And 

The  entire  surface  of  a  cone  consists  of  its  base  and  its 

lateral  surface  (which  is  equal  to  a  triangle  whose  base  is 

the  periphery  of  the  base  of  the  cone,  and  whose  altitude  is 

its  slant  height).  ^ 

°    ^  Process. 

Ex.  1.  The  slant     j  ^  7  sq.  in.  =    4.9  sq.  in.,  area  of  base. 
height  of  a  pyra-  ^      ^^_x^_^  ^  ^^^  sq.in.,area  of  the  i  sides. 
mid  IS  16  inches,  ^  —--^  ^    .  *^ 

and   the   base   is         ^nhre  area,  27S  sq.in. 

1  inches  square.     What  is  the  area  ? 

Ex.  2.  The  periphery  of  the  base  of  a  cone  is  60  inches, 
and  the  slant  height  is  2  feet.  How  many  square  inches  are 
there  on  the  surface  of  the  cone  ? 

Process. 
60  in.  -^  S-f  —  ~Y^',  diameter  of  base, 
(i  of^f  =:)  ^0^-  X  ^-^^  =^     286-/J  sq.  in.,  area  oj  base. 

— — ^^  "''' '"'  =     720  sq.  in.,  area  of  curved  surface. 
Entire  surface  of  cone,  1,300  /j  sq.  in. 

The  number  of  units  in  the  entire  surface  of  a  pyramid 
or  a  cone  is  equal  to  the  sum  of  the  units  in  its  base,  plus 
the  wiits  in  its  lateral  surface. 

Problems. 
Required,  • 

1.  The  surface  of  a  prism  1  ft.  9  in.  long,  and  8  in.  square. 

2.  The  area  of  a  stick  of  timber  50  ft.  long,  and  7  by  10  in. 

3.  The  curved  surface  of  a  granite  column  18  inches  in 
diameter  and  20  feet  high. 

Jf..  The  outer  surface  of  an  open  tank  8  feet  in  diameter 
and  6  feet  deep. 

5.  The  square  yards  on  the  surface  of  a  conical  spire  8  ft. 
4'  in  diameter  at  the  base,  and  72  ft.  6'  high. 


356  SUPPLEMENT. 

Find  the  lateral  surface 

6.  Of  an  octagonal  (or  eight-sided)  spire  6  ft.  on  a  side  at 
the  base,  and  33  ft.  high. 

7.  Of  a  hexagonal  (or  six-sided)  prism  whose  faces  are 
each  3  ft.  by  ^'. 

8.  Of  a  log  28  ft.  long  and  17'  in  diameter. 
Required,  the  entire  area 

9.  Of  a  cone,  the  periphery  being  40  inches,  and  the 
slant  height  3  feet. 

10,  Of  an  octagonal  spire,  each  side  of  the  base  measuring 
5  ft.,  and  the  slant  height  37  ft.  4  in. 

11,  Of  a  cone,  the  diameter  of  the  base  being  1  If  inches, 
and  the  slant  height  2  feet  6  inches. 

12,  Of  a  pyramid,  the  slant  height  being  27  inches,  and 
the  base  11  inches  square. 

Case  II.  To  find  the  volume  of  a  prism,  cylinder, 
pyramid,  or  cone. 

202.  The  volume  in  a  portion  of  a  prism  or  of  a  cylinder 
1  foot  long,  equals  the  number  of  square  units  in  either  base; 
the  volume  of  a  portion  of  the  same  prism  or  cylinder  4,  5, 
or  6  feet  long  is  4,  5,  or  6  times  as  much  as  the  volume  of  a 
portion  of  the  same  prism  or  cylinder  1  foot  long. 

Ex.  The  area  of  the  ^ 

,  n        1  I  Process. 

base    ot    a    hexagonal 

prism  is  78  sq.  ft.,  and  19  ft.  6  in.  ^  19.5  ft, 

its  length  is  19  ft.  6  in.       ^p^^  ^  7^  ^y^p  ^  i^^21  cu.  ft. 

What  is  its  volume  ? 

I.  The  number  of  units  in  the  volume  of  a  prism  or  a  cyl- 
inder is  equal  to  the  product  of  the  number  of  units  in  the 
base,  midtijMed  by  the  number  of  units  in  the  length, 

II.  The  volume  of  a  pyramid  is  one  third  as  much  as  the 
volume  of  a  prism  that  has  the  same  base  and  altitude.    And 
III.  The  volume  of  a  cone  is  07ie  third  as  much  as  the  vol- 
time  of  a  cylinder  that  has  the  same  base  and  cdtitude. 


ME  A  S  UR  EMEN  TS.  367 

Ex.  1.  Tlie  area  of  the  base  of  an  octagonal  pyramid  is 
35  sq.  ft.,  and  its  altitude  is  15  ft.  4  in.    What  is  its  volume  ? 

Ex.  2.  What  Process. 

are    the    cubic  15i  x  S5  cu.ft.  ^^^^^  ^^^^^  ^^^^^^  of  pyramid, 

contents    of    a  ^^ 

cone  6  ft.  in  di-  Process. 

ameter  at  the  Sfx6ft.=l8Yft',  circumference  of  base. 

base,  and    10  §x(^ofl8f  sq.ft.)  =  ^8f  sq.ft.,  area  of  base. 

ft.  Sin.  hish?       10§  x  28  f  cu.ft.      ^^^j,        .,       ,  . 

o  — ^ ^^—z=zl00\  cu.ft.,  wlume  of  cone. 

Problems. 
Required, 

1.  The  volume  of  a  cylindrical  shaft  of  marble  2  ft.  3  in. 
in  diameter,  and  16  ft.  9  in.  high. 

2.  The  cubic  contents  of  a  stick  of  timber  24  ft.  long,  the 
bases  being  right-angled  triangles,  the  two  shorter  sides  of 
each  of  which  are  1 8  in. 

S.  The  capacity  of  a  tube  8  ft.  long  and  4  in.  in  diameter. 
U.  The  contents,  in  gallons,  of  a  cistern  5  feet  in  diame- 
ter, and  6  ft.  deep. 

5.  What  are  the  cubical  contents  of  a  shaft  29  ft.  long, 
and  9'  in  diameter  ? 

6.  What  length  of  wire  -^  of  an  inch  in  diameter  can  be 
drawn  from  a  cubic  foot  of  brass  ? 

7.  How  many  tons  of  hay  can  be  stored  in  a  loft  under  a 
four-sided  roof  30  ft.  square  and  10  ft.  high  to  the  peak? 

In  Case  I,  page  356, 

8.  Find  the  volume  of  the  cone,  problem  9. , 

P.  Find  the  volume  of  the  pyramid,  problem  12. 

10.  In  a  stack  of  hay   19  ft.  in  diameter  and  17  ft.  high, 
with  a  conical  top  17  ft.  high,  are  how  many  tons? 

11.  A  cistern  6i  ft.  in  diameter,  and  8  ft.  deep,  contains 
how  many  gallons  ?     How  many  barrels  ? 

12.  A  rectangular  cistern  12  ft.  long,  10  ft.  wide,  and  9  ft. 
deep  will  hold  how  many  barrels  of  water  ? 


358  SUPPLEMENT. 

Case  III.  To  find  the  area,  and  the  volume  of  a  sphere. 

203,  A  sphere  or  a  globe  is  a  solid  bounded  by 
a  curved  surface,  every  part  of  which  is  equally 
distant  from  the  point  within,  called  its  centre. 

204,  A  hemisphere  is  one  half  of  a  sphere.        Figure  lo. 

205*  I.  The  area  of  a  sphere  is  ^  times  as  much  as  the 
area  of  a  circle  of  the  same  diameter, 

II.  The  volume  of  a  sphere  is  two  thirds  as  much  as  the 
volume  of  a  cylinder  whose  diameter  and  altitude  are^  each^ 
equal  to  the  diameter  of  the  sphere. 

Ex.  Find  the  area  and  the  volume  of  a  13-inch  globe. 
Full  Solution. 

3^  X  IS  in.  r=z  JfO^  in.y  circumference  of  IS-inch  cylinder. 

i  of  IS- 6^,  and  ^  ofi0f  =  20f. 

64^  X  20 f  sq.  in.  =  IS 2f^  sq.  in.y  area  of  base  of  cylinder, 

Jf,  X  1  S2j-^  sq,  in.  =  5314^  sq.  in.,  area  of  y lobe. 
IS  X  lS2'^^cu.in.  —  1^726^j  cu.  in.,  volume  of  cylinder. 
§  of  1 ,7 26 -f J  cu.  in.  =  fl50^^  cu.  in.,  volume  of  globe, 

^       .     ^  Pkoblems. 

Kequired, 

1.  The  area  and  the  vohime  of  a  15-inch  school-globe. 

2.  The  area  of  a  2-inch  ivory  ball. 
S.  The  volume  of  a  10-inch  foot-ball. 

Jf.  The  area  of  a  hemisphere  12  inches  in  diameter. 

5.  The  volume  of  the  earth,  the  diameter  being  7,911  miles. 

6.  What  is  the  area,  and  what  is  the  volume,  of  the  largest 
globe  that  can  be  turned  from  a  block  of  mahogany  which 
is  a  10-inch  cube? 

III.   Dimensions,  Areas,  and  Volumes   of  Similar  Sur- 
faces AND  Similar  Solids. 

206*  Shnilar  surfaces  and  similar  solids  are  such  as 
are  of  the  same  form,  without  regard  to  size. 


ME  A  S  UR  EMEN  T  S,  359 

_,  -  (2  inches,  is  4  square  inches. 

The  area  of  a  square  )  ^  .     i       •    n^  •     i, 

,  .  1    .  •<  5  inches,  is  25  square  inches, 

whose  side  is         \  a  -     \.      -    n  a  •     u 

(  8  inches,  is  64  square  inches. 

The  areas  of  squares  and  other  similar  surfaces  are  to 
each  other  as  the  squares  of  their  sides. 

m         1  c         1     (  2  inches,  is  8  cubic  inches. 

Ihe  vohime  or  a  cube  )      .     ,       .  ,  •    .     , 

T  J      •  -{5  inches,  is  125  cubic  inches, 

whose  edp;e  is         ]      -     ^       -  ,  .    .     , 

(  8  inches,  is  512  cubic  inches. 

The  volumes  of  cubes  and  other  similar  solids  are  to  each 
other  as  the  cubes  of  their  edges. 

Problems. 

1.  A  tract  of  Government  land  160  rods  square  is  a  quarter- 
section.  What  is  the  length  of  one  side  of  a  square  tract 
that  contains  36,000  acres  ? 

2.  If  a  6-inch  cube  of  iron  weighs  60.5  pounds,  what  is 
the  weight  of  a  3-foot  cube  ? 

3.  What  is  the  circumference  of  a  circle  whose  area  is 
3  times  the  area  of  a  circle  20  inches  in  diameter? 

Jf..  The  surface  of  a  globe  1  foot  in  diameter  is  3.1416 
square  feet.  What  is  the  surface  of  a  globe  of  5  feet  in 
diameter  ?     Of  a  globe  of  6  inches  in  diameter  ? 

5.  Find  the  circumference  of  a  globe  whose  volume  is 
64  times  the  volume  of  a  globe  V  inches  in  diameter. 

6.  If  the  volume  of  a  pyramid  20  ft.  high  is  4,600  cu.  ft., 
what  is  the  volume  of  a  similar  pyramid  100  ft.  high  ? 

7.  The  diameter  of  the  base  of  a  certain  cone  is  24  feet ; 
and  from  the  top  part  i  of  its  volume  is  cut  off,  by  a  plane 
parallel  to  its  base.  What  is  the  diameter  of  the  base  of 
the  part  cut  off? 

A  ball  6  inches  in  diameter  weighs  32  pounds; 

8.  What  is  the  weight  of  a  ball  3  inches  in  diameter? 

9.  What  is  the  diameter  of  a  ball  that  weighs  500  pounds  ? 
10.  A  certain  watering-trough,  5  feet  long,  holds  12  pail- 

f uls.     What  is  the  capacity  of  a  similar  trough  8  feet  long  ? 


[360] 


§  31.    FORMS   OF  BUSINESS  PAPER. 

NoTK.— Substitute  other  words  for  the  words  in  Italics,  as  occasion  may  requlra 

I.  Promissory  Notes. 

On  Demand. 

t^S^Mf  Chicago,  Sept.  15, 1882. 

On  demand  I  promise  to"  pay  to  Joseph  Young,  or  order,  Two  Hun- 
dred Thirty-five  and  ^q-q  Dollars,  with  interest,  for  value  received. 

Jai.  T.  Bryant. 

Time,  without  Interest. 
$60  Detroit,  Dec.  1, 1882. 

Six  months  after  date,  I  promise  to  pay  to  John  Smith,  or  order,  at 
his  office,  Sixty  Dollars,  for  value  received.  p^^^  j^^^ 

Time,  with  Interest. 
$56^^  Cleveland,  Oct.  3, 1882. 

Ninety  days  after  date,  I  promise  to  pay  to  Wm.  Brown,  or  bearer, 
Fifty -six  and  /^V  Dollars,  with  interest,  for  value  received. 

David  Greene. 

Bankable  Note. 
%lflOO  Boston,  June  21,  1882. 

Two  months  after  date,  I  promise  to  pay  to  the  order  of  Edward 
Eliot,  at  the  Fourth  National  Bank,  One  Tlwusand  Dollars,  for  value 
received.  ^^^^^  Hudson. 

Joint  Note. 
%290  St.  Louis,  Jan.  20, 1882. 

Sixty  days  after  date,  we  promise  to  pay  to  Henry  W.  Raymond,  or 
order,  Two  Hundred  Ninety  Dollars,  for  value  received. 

Davis  &  Williams. 
Merchandise  Note. 
$750  Milwaukee,  Feb.  Jf,  1882. 

On  the  first  day  of  August  next,  I  promise  to  pay  to  Ray  &  Mitchsll, 
or  order,  at  their  mills  in  this  City,  Seven  Hundred  Fifty  Dollars,  in  No. 
1  wJieat,  at  the  market  price  when  due,  for  value  received. 

Edwd.  Denton. 


FORMS   OF  BUSINESS  PAPERS.  361 

Judgment  Note. 
$iOO  Pittsburgh,  March  17, 1882. 

One  year  after  date,  I  promise  to  pay  to  Joseph  Home  &  Co.,  Four 
Hundred  Dollars,  with  interest,  for  value  received. 

And  I  do  hereby  authorize  any  attorney  of  any  Court  of  Record  in 
Pennsylvania  or  elsewhere,  at  any  time  after  the  above  promissory 
note  becomes  due  and  remains  unpaid,  to  confess  judgment  for  the 
above  sum,  with  release  of  errors. 

Witness  my  hand  and  seal,  the  date  above  written.  ,^j.^ 

Witness  present,  Abel  N.  Brown.  ]  L.S.  >■ 

Chas.  Clark.  ^  ^-^  ^ 


II.  Due -Bills. 
Payable  in  Money. 
$15  Buffalo,  July  1, 1883. 

Due  Sylvia  J.  Eastman  Fifteen  Dollars,  one  day  after  date,  for  value 
received.  ^^^^  Stoneman. 

Payable  in  Merchandise. 
$^^  -f-Q-u  Portland,  Nov.  29, 1882. 

Due  W.  J.  Hunt,  or  bearer,  Ninety-six  Dollars  and  Fifty  Cents,  pay- 
able in  goods  from  our  store.  Johnson,  Luce  &  Co. 


III.  Orders. 
Payable  in  Money. 

, ,    -, ,     ^  Hartford,  May  17,  1882. 

Mr.  Peter  Cooper,  j      ^       u      , 

Please  pay  to  Henry  H.  Rice  Ten  Dollars,  on  my  account. 

Oliver  Turner. 

Payable  in  Merchandise. 

Burlington,  March  18,  1882. 
A.  J.  Mason  &  Co., 

Please  pay  to  Andrew  Adams,  or  bearer.  Sixteen  Dollars  and  Twenty 
five  Cents  ($16y%^^),  merchandise,  and  charge  to  my  account. 

B.  F.  Baker. 
Bank  Check. 
$210^^0^  Springfield,  Sept.  1, 1882. 

THIR33    T>r^TI01S^AL.    BANK. 

Pay  to  E.  A.  Hubbard,  or  order,  Two  Hundred  Ten  and  //^r  Dollars. 

Wood  cfc  Harrison. 

Q 


362  SUPPLEMENT. 

IV.  Drafts. 

Sight  Draft. 
$Jf93-fjl>^  Omaha,  Neb.,  Aug.  U,  1882. 

At  sight,  without  grace,  pay  to  the  order  of  Day  &  Martin,  Four 
Hundred  Niuety-three  and  7%  Dollars,  value  received,  and  charge 
to  our  acc't.  p^^^^  ^  j^^^^ 

To  C.  C.  Gould, 

St.  Paul,  Minn. 

Time  Draft. 
$000  Denver,  Col,  Feb.  10, 1882. 

Ten  days  after  date  pay  to  Edicard  Newton,  or  order.  Nine  Hundred 
Dollars,  for  value  received,  and  charge  to  acc't  of 

To  C.  S.  Griggs  d:  Co.,  G.  W.  Sumner. 

Chicago,  Til. 

Time  Draft,  collectible  through  a  Bank. 
$3,500  Philadelphia,  Jan.  ">,  1882. 

Ten  days  after  sight,  pay  to  the  order  ol  George  II.  Logan,  at  the 
Chemical  Bank,  Three  Thousand  Five  Hundred  Dollars,  and  charge  to 
acc't  of  TJiomas  IlunUr, 

To  D.  Appleton  d;  Co.,  716  Filbert  Street. 

New  York  City. 

V.  Receipts. 

In  Full  of  all  Demands. 
J  ^  ^  T%  Albany,  June  27,  1882. 

Received  of  E.  H.  Hansom  &  Co.,  Ninety -seven  and  7%  Dollars,  in 
full  of  all  demands.  s^  2^.  Gray. 

By  Agent  or  Clerk,-in  Full  of  Accounts. 
ZZZ-L      Received,  Cincinnati,  January  20,  1882,  of  Wm.  T.  Harris, 
Five  Hundred  Dollars,  in  full  of  accounts  to  date. 

David  Pickering, 

per  Henry  Wood, 

For  Note  on  Account. 
$89 6^^  New  York,  May  4, 1882. 

Received  of  Anson  Bichniond  of  Troy,  his  note  at  three  months,  dated 
2d  inst.,  for  Three  Hundred  Ninety-six  and  //^  Dollars,  on  account. 

W.  &  B.  Douglas, 

87  John  St 


[363] 

\  33.  TAX,  COMPOUND  INTEREST,  AND  COIN  TABLES. 
I.  Tax  Table. 


(Rate  of  valuation,  1^ 

mills  on  a 

dolUr,  or 

.15%  of  assured  valus 

Ltion.) 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax.      1 

Prop. 

Tax. 

$1 

$.0015 

$10 

$.015 

$100 

$  .15 

$1,000 

$1.50 

2 

.0030 

20 

.030 

200 

.30 

2,000 

3.00 

3 

.0045 

30 

.045 

300 

.45 

3,000 

4.50 

4 

.0060 

40 

.060 

400 

.60 

4,000 

6.00 

5 

.0075 

50 

.075 

500 

.75 

5,000 

7.50 

6 

.0090 

60 

.090 

600 

.90 

6,000 

9.00 

7 

.0105 

70 

.105 

700 

1.05 

7,000 

10.50 

8 

.0120 

80 

.120 

800 

1.20 

8,000 

12.00 

9 

.0135 

90 

.135 

900 

1.35 

9,000 

13.50 

Ex.  By  tlie  above  table,  find  the  tax  of  ^f^^w  $5,000, 
a  person  who  is  assessed  on  $5,642  ;  and  «  "  uo\ 
who  pays  for  3  polls,  at  $.62^  each.  5  poUs] 

Total  tax, 

XL  Compound  Interest  Table, 


Sliowing  the  amount  of  $1  at  2i,  3,  31-,  4,  5,  6,  7,  and  8  per  cent  compound  interest  for 

any  number  of  years  from  1  to  20 

Years. 

2X  per  ct. 

3  per  cent. 

3%  per  ct. 

4  per  cent. 

5  per  cent. 

6  per  cent. 

T  per  cent. 

8  per  cent. 

1 

1.025000 

1.030000 

1.035000 

1.040000 

1.050000 

1.060000 

1.070000 

1.080000 

2 

1.050625 

1.060900 

1.071225 

1.081600 

1.102500 

1.123600 

1.144900 

1.166400 

3 

1.076891 

1.092727 

1.108718 

1.124864 

1.157625 

1.191016 

1.225043 

1.259712 

4 

1.103813 

1.125509 

1.147523 

1.169859 

1.215506 

1.262477 

1.310796 

1.360489 

5 

1.131408 

1.159274 

1.187686 

1.216653 

1.276282 

1.338226 

1.402552 

1.469328 

6 

1.159698 

1.194052 

1.229255 

1.265319 

1.340096 

1.418519 

1.500730 

1.586874 

7 

1.188686 

1.229874 

1.272279 

1 .315932 

1.407100 

1.503630 

1.605782 

1.713824 

8 

1.218403 

1.266770 

1.316809 

1.368569 

1.477455 

1.593848 

1.718186 

1.850930 

9 

1.248863 

1.304773 

1.362897 

1.423312 

1.551328 

1.689479 

1.838459 

1.999005 

10 

1.280085 

1.343916 

1.410599 

1.480244 

1.628885 

1.790848 

1.967151 

2.158925 

11 

1.312087 

1.384234 

1.459970 

1.539454 

1.710339 

1.898299 

2.104852 

2.331639 

12 

1.314889 

1.425761 

1.511069 

1.601032 

1.795856 

2.012197 

2.252192 

2.518170 

13 

1.378511 

1.468534 

1.563956 

1.665074 

1.885649 

2.132928 

2.409845 

2.719624 

14 

1.412974 

1.512590 

1.618695 

1.731676 

1.979932 

2.260904 

2.578534 

2.937194 

15 

1.448298 

1.557967 

1.675349 

1.800944 

2.078928 

2.396558 

2.759032 

3.172169 

16 

1.484506 

1.604706 

1.733986 

1.872981 

2.182875 

2.540352 

2.952164 

3.425943 

17 

1.521618 

1.652848 

1.794676 

1.947901 

2.292018 

2.69^773 

3.158815 

3.700018 

18 

1  559659 

1.702433 

1.857489 

2.025817 

2.406619 

2.854339 

3.379932 

3.996020 

19 

1.598650 

1.753506 

1.922501 

2.106849 

2.526950 

3.025600 

3.616528 

4.315701 

20 

1 .638616 

1.806111 

1.989789 

2.191138 

2.653298 

3.207136 

3.869085 

4.660957 

364 


SUPPLEMENT. 


Use  the  compound  interest  table  on  page  363  as  follows : 
I.  To  find  the  amount  of  any  given  principal  for  years  -.—Multiply 
the  amount  of  $1  for  the  given  number  of  years  at  tlie  given  rate  ^,  by 
the  number  expressing  tlie  principal. 

II.  To  find  the  compound  interest  '.^Subtract  the  principal  from  the 
amount. 

When  there  are  months  and  days  in  the  given  time  i—Find  the  sim^ 
pie  interest  on  the  amount  for  the  montJis  and  days,  and  add  it  to  the 
amount  for  the  years.     The  sum  will  be  t?ie  amount  for  the  whole  time. 

III.  Estimate  of  Values  of  Foreign  Coins. 
As  proclaimed  by  the  Secretary  of  the  Treasury,  January  2, 1882. 


Country. 

Monetary  Unit. 

Standard. 

Value 

1  u.  s. 

Money. 

Austria                     

Florin 

Silver 

$  .10,6 
.19,3 
.82,3 
.54,6 

1.00 
.91,2 
.93,2 
.26,8 
.82,3 
.04,9 
.19,3 

AMM 
.19,3 
.23,8 
.96,5 
.39 
.19,3 
.88,7 

1.00 
.89,4 
.40,2 
.26,8 
.82,3 

1.08 
.65,8 

1.00 
.19,3 
.26,8 
.19,3 
.74,3 
.04,4 
.82.3 
.19,3 

Belffiiiin    . .               .... 

Franc 

Boliviano 

Gold  and  silver.... 
Silver 

Bolivia   

Brazil    

Milreis  of  1,000  reis  .... 
Dollar 

Gold 

British  Possess,  in  N.  A... 
Chili     . . 

Gold 

Peso 

Gold  and  silver 

Gold  and  silver.... 
Gold 

Cuba 

Peso 

Denmark 

Ecuador 

Crown 

Peso 

Silver 

EervDt 

Piaster 

Gold 

Gold  and  silver .... 
Gold 

France  

Franc 

Pound  sterling 

Great  Britain 

Greece 

Drachma 

Gold  and  silver 

Gold 

German  Empire . . 

Mark 

Hayti       

Gourde 

Gold  and  silver .... 
Silver 

India 

Rupee  of  16  annas 

Lira 

Italy 

Gold  and  silver .... 
Silver 

Japan 

Liberia 

Yen 

Dollar 

Gold 

Mexico 

Dollar      

Silver 

Gold  and  silver.... 

Gold 

Silver 

Netherlands 

Florin 

Norway 

Crown 

Peru 

Sol 

Portugal 

Milreis  of  1,000  reis.... 
Rouble  of  100  copecks. 
Dollar 

Gold 

Russia 

Silver 

Gold 

Sandwich  Islands 

Spain  

Peseta  of  100  centimes. 
Crown 

Gold  and  silver 

Gold 

Sweden 

Switzerland 

Tripoh    

Franc 

Gold  and  silver.... 
Silver 

Mahbub  of  20  piasters. 
Piaster 

Turkey 

Gold 

U.  S.  of  Colombia 

Venezuela 

Peso 

Bolivar 

Silver 

Gold  and  silver 

[365] 


ANSWERS  TO   WRITTEN   PROBLEMS. 


In  most  of  the  answers  expressing  United  States  money,  ivhen  the  mills  are  5  or 
more,  they  are  regarded  as  1  cent ;  and  when  less  than  5,  they  are  rejected. 


INTEGERS.  Addition.  Art.  39.  A.  1—19.  2—19.  3—2^. 
^_15.  5—14.  6—19.  7—26.  <?— 25.  S*— 23.  10—19.  11—20.  12—19. 
13—21.  U—1^.  15—21.  16—19.  i7— 32.  18— 2b.  B.  1—4:2.  2-26. 
,?— 29.  ^—33.  5—41.  6—30.  7—22.  8—25.  P— 31.  10—27.  11—30. 
12—4:0.  13—21.  U—39.  15— U.  16—30.  17—20.  18-23.  19—29. 
20—32.  21—24.  22—43.  23—33.  C.  1—102.  2—10.  3—94,.  ^—103. 
5—92.  6—113.  7—74.  ^—88.  5—86.  it*— 118.  ii— 89.  i^— 68. 
13—04.  U—12.  15— '62.  16—^0.  17—07.  18-78.  19—00.  20—79. 
21—00.  22—75.  23—77.  24—73.  25-81.  26—84:.  27—05. 
■  Art.  45.  A.  i— 2,070.  ^—$10,688.  ^—12,053.  4— $110.84. 
•5—9,666.  6— $1,940,554.  7—2,673,757.  5— $9,952.23.  5—2,026,125. 
B.  i— $166.  ^—$89.  ^—$105.  4— $133.  5— $77.  6— $90.  7— $131. 
^—$82.  5— $113.  i^— 1,731.  ii— 1,521.  i^— 1,830.  i^— 1,748. 
i^— 899.  i5— 946.  i6— 2,325.  i7— 1,416.  i<§— 1,244.  iP— $170.35. 
^5— $102.09.  ;^i— $181.83.  ;^^— $249.02.  ^^—$301.89.  ^^—$60.83. 
;^5— $258.99.  ^6— $138.04.  ^7— $138.66.  ^<?— $119.99.  ^5— $242.79. 
30—$107M.  n.  i— 51,360.  ;^— 3,399,395.  5— $94,412.74.  4— $11,975.23. 
5—47,311,999.  6—24,793,183,104.  7— $430,421.05.  F.  i— 3,366  lb. 
;^— $514.58.  ^—$294.27.  ^—2,143  bu.  5— 6441b.  6— 2111b.  7— $26.38. 
^—957  cd.     5— $681.     i5— $6,484.     ii— 7,338  logs,     i^— $420. 

Art.  46.  ^.  i— 13,790.  ;^— 1,939.  .5— $5,410.  ^—29,205.  5— $8.87. 
6—14,096.  7— $224.  <?— 276  persons.  5—282  barrels.  10—134  rd, 
ii— 694  mi.  i^— $2,405.  i^— 1,989  bu.  C.  i— $17,995.  i"— $27,918, 
^—36,816,218.  .4— $8,156.  5— $80.10.  6— $5,842.  7— $20,855, 
<^— $444.92.  5— $2,810.44.  i^— 270,011,432.  ii— 48,637,000  sq.  mi 
^^— 3,784  cd.  iJ-- 6,6491b.  i^— $28,216,103.  i5— $743.42.  i6— $1,660.44. 
i7— $7,144.44.  i^— $1,110.90.  i5— 3,198  bushels.  ^^—$1,836.79, 
^i— 69,515,137.     ;^^— 939,626,817. 

Subtraction.  Art.  59.  A.  1—432.  2—411.  3—430.  .4—374. 
5—367.  6—63.  7—223  pins.  <9— 708  soldiers.  5—2,485  shimrles. 
i5— 15,992  pounds,  ii— $14,441.  i^— $3,804.98.  i^— $37,126.44. 
i^— $5,626.75.  E.  i— 221,208.  ^—1,694,178.  .?— 74,493,314. 
.4—167,025,150.  5—4,987.  6— $6,128.  7—36,908.  <?— $13.59. 
5—968, 724.  10—7, 348, 889.  ii— 253, 292, 037.  i^— $15, 013. 19. 
13— $385.79.  6?.  i— 24,112  T.  ^—282  baskets.  ^—$777.  ^—$1,464. 
5—5,351  children.  6— $9,376.  7— $814.76.  <§— $6,080.81.  5— $1,880. 
i6'— $2,272.  ii— 14,513  bu.  i^— $46,792.  i^— 259  A.  i^- 2,023  cd. 
i5— 19,419  ft.  i6— 1,669  lb.  i7— 24,484.  i<^— 5,373  square  milea 
i5— 3,247  lb. 


366  ANSWERS    TO    WE  IT  TEN-  PROBLEMS. 

Art.  60.  A.  i— 863.  ^— 1373,695.  5—68,894,728.  4— 9,110,998,17a 
5—15,624.39.  6— $2,088.22.  1^.  i— 386.  ^—568.  .?— 48,023.  4—182. 
^—47,637.  67—47,455.  7—938.  ^—1,990.  5—10,200.  /^— 1,052. 
ii— 9,262.  i^— 8,210.  7J— 435.  74—9,020.  i.5— 100,045.  7^—8.585. 
i7— 99,610.  i<^— 91,025.  i9— $953.25.  ^^—$4,758.25.  fi— $98,466.25. 
;^^— $3,805.  ^J— $97,513.  ;^4— $93,708.  ;^5— $8.86.  ;^6'— $64j02. 
J^7— $180.58.  ^5— $55.16.  ;^5— $171.72.  ,?^— $116.56.  5i— 1,998. 
5-^—15,001,986.  55—99,073.  54—4,164.  J),  i— 1,728.  ^—1,732. 
5—48  yr.  4—55  yr.  5—211  yr.  6—228  yr.  5— $5,320.  P— $10,761. 
i^— 15,668  ft.  ii— 8,285  riflea  i^— 116,664,982.  i5— 12,863,002  bu. 
i4— $16,156. 

Bemeio.  i— $1,155.  J^— 7,018  persons.  5— $5,744.  4—44,976,487. 
5— $449.  6— $3,056.  7— $10,031.19.  5—175  hhd.;  176,235  lb. 
P— 390,273  T.     iO— 1,492  Gopies. 

Multiplication.  Art.  7>$.  A.  i— 738.  ;?— 3,801.  5—16,345. 
4—76,992.  5—75,728.  (7- $3,928.  7— $32,562.  5— $28,228.  5— $544.70. 
i^— $17,661.60.  /i— 142,492  yd.  i^^— 771,024  ft.  i5— 216,188  lb. 
74—360,063  T.  B.  i— $2,275.  <^— $76,050.  5— $12.30.  4— $976. 
5— 840  1b.  6— 42,240  ft.  7— $51,072.  5— 221,160  gal.  £>— 160,255  lb. 
i6>— 6,158,088.      77- $105;   $112.50.      7;?— $44.38. 

Art.  77.  7—7,500.  -^—75,000.  5—39,200.  4—1,839,000.  5—3,920,000. 
6—28,930,000.  7—478,000,000.  <9— 58,0.54,000,000.  9—45,360. 
i^— 272, 800.  77—13, 356, 000.  7  J"- 625, 230, 000.  75—1 2, 1 78, 600, 000. 
i4— 450,430,000,000.  75—7,620.  76— 996,000;  996,000,000.  77—583,200; 
583,200,000.  75—194,850,000.  79—146,400  bushels.  ^(9—29,400  lb. 
;^7— 294,000  gallons.  :e^— 42,800  lb.  eS— 1,440  min.  ^4— $19,500. 
I?5— $2,500.  ;^6— $8,800.  ;^7— $87.50.  -^5— $350.  ;^9— $4,860. 
59-197.50.      57— $2,550. 

Art.  80.  A.  7—1,702.  ^—15,174.  5—241,056.  4—134,190. 
5—1,222,904.  6—6,139,776.  7—48,727,272.  5—3,845,355.  9—149,598,976. 
i(9— 48,078,027.  77—9,106.  7;?— 43,289;  155,318,240.  75—11,702,718. 
i4— 4,314,156  yd.  75—2,368  lb.  i6— 1,044  eggs.  77—72,592  axes. 
75—66,836  rm.  79—67,482  lb.  B.  7— $1,118,700.  ;^— $11,187. 
5— $35, 560. 20.  4— $52, 406. 25.  5— $10, 252, 602. 23.  6— $6, 000. 
7— $225,952.  5— $2,112.  9— $77,698.25.  79— $109,515.  77— $1,267.36. 
i^— $7.56.  75— $8.68.  i4— $236.16.  75— $70.08.  76— $6.21. 
77— $3,996.75. 

Art.  82.  7—403,200.  ^—68,640.  5—259,200.  4—1,248,000,000.' 
5—308,160,000.  6—3,360,000.  7—7,822,118,707,200.  5—2,623,023,200,000. 
9—183,230,150,040.  79— 2,331,6001b.  77— 42,000  words.  7^— $20,400. 
i5— $120,000.  74—962,000  feet.  75—174,000  lb.  76—51,200  poles. 
77— $476.  75— $125.  79— $12,375.  ^9—7,400  lb.  f7— 43,240  lb. 
^^—124,800  sheets.    ^5— $850,000.    ^4- $325,000.    ;^5— 3,895,500,000. 

Art.  83.  A.  7—6,158,088.  f— 326,442.  5—793,032.  4—70,376,400,000. 
5—210,457,200.  6—4,480.000.  7—4,234,000.  5—19,800,000. 

9—116,550,000.      79—266,486,552.      77—756  wagons.      7^*- 1,344  A. 
75—136,300  lb.      74—8,400  barrels.      75—34,823  T.      76— $232,650. 


AjVSWUBS  to  written  problems.       367 

i7— 25,273  persons.  ^^—642,000  ft.  C.  i— 55,615;  490,609.  ^—186,800 
1,500,000.  <?— 39,474;  3,508,800;  16,508,904.  ^—70,641,120 
1, 881 ,  369, 600.  5—59, 373.  6—892, 367, 000.  7—27, 618, 720, 000. 
^—29,836,064,832.  P— 14,946,816.  i6'— $100,608,000.  ii— $56,068.74. 
i^— $111,457,304.  i^— 1,107,087,782,400.  i^— $6,589,624,320. 

i5— 258,720  lb.  i6— 56,940  times.  i7— 22,908  eggs.  i<?— 631,800  mi, 
i9— 21,600  sheets.  ^6>— 30,240  buttons,  ^i— $1,620.  ^^—81,765  bar. 
^«?— $1,244.10. 

Bevieio.  i— 900  sq.  mi.  ;^— 5,040  lb.  ^—$322.  ^—$64.  5- 
6—2,046  pickets.  7— $1,045.25.  ^—$11,232.  P— 289,085  pounds, 
i^— 18,648  yd.  ii— $200.  i^— 1,462  bu.  i^— $58,500.  i^— 686  bu 
jr5— 5,3001b.  i6— 89  sets;  5,033  pieces.  i7— $568. 50 gain,  i^— $15,146. 
19— B,  $1,350;   C,  $8,100;  All,  $10,125. 

Division.  Art.  97.  i— 232.  ^-212.  <?— 213.  .4—411.  5—412. 
6—206.     7—420  papers.     5—210  bricks.     9—506. 

Art.  99.  i— 821  da.  ;^— 401  da.  ^—$4,688.  .4— $6,525  received; 
$4,350  paid.  5— $1,208.  6— 205  A.  7— $1,236.  <9— 251  lb.  5— $8,046. 
ii— 948  bar.  i.^— 339  da.  i^— 243  da.  i^— 37mi.  i5— $13.  i6— $18. 
i7— 184  bu.  18—^  cars.  i9— $48,396.  20—10^  days,  .^i— 16  oz. 
:^.^— 275  sacks.  23— %4:.  ^.4— $33.  ^5—5  farms,  18  A.  remainder. 
;^6— $1,005.  .^7—18  full  cargoes.  28—212  bales,  48  lb.  remainder. 
;^5>_$5.37.  30— 12^)  pens,  ^i— 18  mo.  ^^—23  sets.  ^^—$136.99. 
^4—16  cars.  35—mi^i-,\.  36—1Q4:  dozen  pairs.  «?7— 168  boxes. 
38—24:  bells.     .55—40,508.     .4^—327. 

Art.  102.    i— 6,475.    .^—1,627.    3—1^^-^^-^.    ^—32,470.    5—7,250. 

6— 3,2963:VVo'o-  7— $4.30.  5— $.08.  9— $26.25.  iC*— $3.75.  ii— $18.50. 
12—%.m.    i.?— 18 horses,  $7 left,    i^— 375 bonds.    i5— 583 freight-cars. 

Art.  104.  i— 15  steam-tugs.  ^—192  thousand  ft.  ,5-7  payments  ; 
last  payment,  $156,750.  .4— 157ff  bu.  5—98  full  reams.  6—48  organs. 
7— 43  chests.  5— 24  hours.  5— 86^Wo-  10—in^ii^%  ii— 22||tf||. 
i^-365,482#oHf(fo-  13-12mro-  i^-2,102|ffif.  i5-143ifaif^. 
16—10^.    17— $26.    i^*— Quotient,  2,102 ;  rem.,  39,875.    i5— 376  horses. 

Art.  105.  A.  i— 831.  .^—912,566.  ,5- 2,421f f.  ^—8,769.  5—372-,^^%. 
6_399iioo  7_i^450.  ^_l,143ff|.  5— l,941fff.  ii— 546| ;  409 1 ; 
468f.  if— 4,122;  2,748;  2,061.  i.?— 125  ;  327.  i^— If  §§  ;  Slfilt; 
nUl  i5— $1.01;  $4.03;  $19.70.  16—Qj%%;  320x1^;  436ii|.  i7— 162. 
i5— $152.60.  i9— 288.  f^— 108.  21-41^.  22—lQ^%.  23—51{^. 
^^_|4.84.  ^5-4,400.  ^6- 961|t^^.  ^7-9,236.  f<S'— 3,273. 
^.  i_4,059  sheep,  f— 393  bar.  .5-192  mi.  4—450  A.  5—3  dress 
patterns,  4  yd.  rem.  6—48  machines.  7—44  da.  5— $2.25.  9—$  .15. 
i^_500  oranges,  ii— 1,188  bar.  if —209  da.  i.5— $134.  i^— 70  bu. 
i5— 4,124  bu.  i6— 23  full  casks.  i7— $17.  i<5— 60  lb.  i9— 46  mi, 
20—20  quires,      f  i— 57  kettles,  164  lb.  rem. 

Art.  106.  i— 217.  f— 510.  .5—229.  4— 353.  5—254.  6—518. 
7—9  yr.  5—11  in.  9—31  miles.  i(9— 135  gallons,  ii— 137  pupils. 
12—^6  yd.     i.5— $31.     i.f-$333.08. 


368  ANSWERS   TO    WRITTEN   PROBLEMS. 

Art.  107.  i— 66,880.  ;e— 8,634,504.  <?— $675.91.  ^—260,618,400. 
5—28,784.  C— 4,663,400.  7—29,798.  S—U.  9—27,012.  10— 2m. 
ii— 248,400.  i^— 666,639.  iJ— $1,180.  i^— 640.  i5— 4,680,000. 
iG— 7,000.     i7— 2,297.     i<^— 36.     19— 15,22o.     <^^— 1,153. 

Review  Problems  in  Integers.  ^.  i— $83. 33 J;  $19. 235V  '^— 4,855  bu. 
^—$7,020.  ^—8,102  bar.  J— $75,183.  6—2,681  bales,  112  lb.  rem. 
7— 96 mi.  <?— $244.  5— 1,405yd.  10— 2,000\h.  ii— 3yd.  i^— $4,427. 
i^— 1,176  T.  i4— 390  lb.  i5— 14  payments  ;  last  payment,  $975. 
16— SO  tubs.  B.  i— 116  sheep.  ;^— $4.80.  3—22  T.  4— $18.24. 
5— $3  ;  $4.  C— 640  A.  7—109  head.  ,?— $123.  9—8  payments. 
10— $7.  ii— $56.  i^— 41  dress  patterns.  i<3f— 1,044  cd.  i^- Cap- 
tain, $980;  mate,  $420;  each  sailor,  $70.  i5— $36,150.  i6— Gamed 
$578.     i7— $4,348.     i«^— $9,797.75.     i5— $669.25. 

DECIMALS.    Addition.   Art.  124.  i— 968.59254.  ^—13. 88875  T. 

5_3,324.35.     .^—$1,645.26.     5— $294.27.     6— $1.41.  7— $2,413,425. 

5— 261. 033  da.    5— 24  oz.    iO— 2.839  T.    ii— 40.05  A.  i^— 25.125  bu.' 

i,^— 14.725  bu.       i^— 119,024.297458.      i5— $247.95.  i6— $310,255. 
i7— $14.33.     i<9— 737.1925  gal.     ii>— 2,460.40072. 

Subtraction.  Art.  125.  i— 125.92.  ;^— 77.166.  .^—1.02955. 
>^— 373.875.  5—19,517.76744.  (>— $62.85.  7—$  .017.  ^—$94,125. 
^—$.935.  iO— $47,125.  ii— 16.7  yd.  i-^— 281.4066  mi.  i^— 1.66  in. 
14—72A5  miles.  i5— 10.75  gallons.  i6— 68.13  rd.  i7— 2.9375  lb. 
i<^— 85.3605.  i9— 75.25  feet.  ^0—S71.25  grains.  ;^i— 840.94946. 
j^^?_753.1875  lb.  ^J— $.68|.  ;^^— 127.6875  barrels.  ;g'5— $61.25. 
^6—^Q7.Q2.  ^7—314.485  tons,  f^— $3,875.  <^9— 16.09375  bushels. 
^^_|2.40.  ^i— 431.89T.  ,?^— 34.55  gal.  33—$lS.22.  J^— 55.266  A. 
,^5— $272.20,  gain  on  oats;  $45,125,  gain  on  corn;  $317,325,  loss  on 
wheat ;  neither  gain  nor  loss  on  all. 

Multiplication.  Art.  126.  i— 1,896.615.  ^—48,187.04.  «^— 20.655. 
^—17,526.25.  5—348,572.16.  6— $3,495.06.  7— $73,337.  <9— $37,830. 
5— $6,240.  i^— $1,275,493.45.  ii— 1.1583.  i^— 523,980.93.  13— 
$12,963.63.  i.4— $19,342.25.  i5— 4.013380638.  i6— 82.5  ft.  i7— 83  bu. 
i<5— 3,907.5  days  (2  leap-years).  i£>— 1,791.875  lb.  -^C*— $1,045.25. 
^i_|12.76.  -^^—$7.50.  f J— $573.50.  -^4— $6,525.40.  ^5— 284. 64426  T. 
^6—1.59494  T.     ;^7— 785.925  lb.     ^,§—1,892.2725  gal. 

Art.  127.  i— .0027328.  j^— .00588.  ,?— .00002358.  .4— .00002793. 
5_. 00399432.  6— .015  bushel.  7— .09  mile.  ^—$.07.  5- $.08. 
i^_.0279  gal.      ii— .000224. 

Art.  128.  i— 537.8.  ^—$23.10.  ^—687.94.  4—59,604.3.  5— $6.25. 
e— $378,125.     7— .62  mi.     ^—$42,697.50.     £'—$75.     it*- $712.50. 

Art.  129.  i— 8,642.  ^—$23,478.  ^—12.007655.  4—$  .80. 
5— .00000496.  6—16.  7—151,136,355.78.  <?— $587,500.  9—625. 
i^— 49.  ii— .765  pound.  i^— 1,568.75  yards.  i^— 472.32  rods, 
i^- 15,472.35  T.  i5— 144.375  gallons.  i6— $3.57.  i7— $15,937.50. 
i<§— $57.50.  i9— $102,375.  ^C*- $13.04.  ^i— $100.  ^^—$18.40. 
;^.^— $8.10.     ^.4— $10,625. 


ANSWERS    TO    WRITTEN  PROBLEMS.  369 

Division.  Art.  130,  i— .044.  ^—.023.  <^— .608.  4— .07517. 
5— ,036  T.      6— .3125  bar.      7— $2;  25.     ,5— $2.25.     5/— .625  bu. 

Art.  131.  i— 37.  2—m.  .?— 15bar.  4— 23  sheets.  5— 19  over- 
coats,    e— 273  casks.     7—17  loads.     5—319  lb.     9—Al  A. 

Art.  132,  i— 1,402.5.  f— 140.25.  .^—14.025.  .^—1.4025.  5—14025. 
e— .014025.  7—1.4025.  5—14.025.  S*- 140.25.  ii— .421+.  i^— 1:^5.399+. 
i^— 12.076+.  i^— $4,875+.  i5— 9.24oz.  i6— 40.5A.  i7— 54.375  bu. 
18—%  .337.     i5— $4,531,892  +  . 

Art.  133.  i— 130.03  +  .  ^—280.  ^—13  sacks.  ^— 6.4h.  5— 561b. 
6—14  bottles.     7— .03125  gal. 

Art.  134.  i— .023.  ^—.0625.  ,^—.00625.  .^—.003125.  5— .00546875. 
6— .037664+.     7— .0053.     5—$  .04.     £/— .0095  oz.     i^— $  .009. 

Art.  135.    i— 53.78.    ^—1.25.    .f— 6.6227.    .^—$.0075.    5— $.4375. 

6—2.06288.    7— 43.45  A.    5— 54.375  bu.    £>— 9.375  mo.    iC*- $41,319.30. 

Art.  136.  j'— .048.  ^—18,400.  5— .078125.  4— .00625.  5— 56.8cd. 
6—40.5  times.  7—422.4  lb.  5—10.88  li.  5— .512  h.  it*- 65.625  bu. 
ii— 625.5  gallons.  i-^— 297.5  bushels.  i5— S5.375.  i^— 280  lb. 
i5— 23  lounges.  i6— $1,875.  i7— .75  bu.  i5— 43.5  yd.  19—2^  gob- 
lets, 3.5  oz.  rem.  ^6>— 407  T. ;  1,105  lb.  over.  1^^-13  full  car  loads, 
7  T.  over.  £2—5S  sheep,  1.5  bushels  rem.  S3— 7  lots,  .075  A.  rem. 
^.4—35  da.,  3  lb.  over. 

Art.  139.    ^—$6.13.     ^—$173.25.     4— $244.50.     5— $40.44. 

RevieiD.  1—2 A.  ^—99,990.  5— .00078125.  4— $15.38+.  5— 14,525.28  T. 
6— $1,142.10.  7— $18.25.  5— $1.50  for  each.  9— .125  bu.  i6>— .475  A. 
ii— $5,234.90.  i^— 66  cans  ;  25  pounds  left.  i5— 58.625  bushels, 
i^— $15,746.71.  i5— $18,375.  i6'— $4,375.  i7— 1,357.05  bushels. 
i5— 1.5946  T.  iP— .6875  of  the  vessel,  ^t*— Lost  $712,875.  ^i— $18.28+. 
;^^— $40.11.     ^5— $92.50. 

PROPERTIES  OF  NUMBERS.  Factors  and  Divisors.  Art. 
144.  i— 3,  3,  5,  7.  2—2,  2,  109.  5—3,  5,  37.  4—863. 
5—3,  3,  3,  3,  3,  3.     6—2,  3,  3,  53.     7—2,  3,  11,  43.     5—3,  3,  5,  7,  31. 

Art.  145.  1—2,  2.  ^—2,  2,  2,  2,  2.  5—2,  2,  2,  3.  4— 2,  2,  2,  5. 
5—3,  5.      6—2,  2. 

Art.  149.  i— 32.  ^—240.  5—28.  .^—51.  5—351.  6—15.  7—72. 
5—28.     P— 14.     iP— 405.- 

Art.  150.  i— 14.  ^—4.  5—28.  4—18.  5—9.  6—6.  7—5. 
5—450.  9—20  rd.  10—2  bu.  at  a  load  ;  600  loads  of  wheat ;  433 
loads  of  corn  ;  392  loads  of  barley ;  893  loads  of  oats.  11—40  yd. ; 
65  rooms.      12 — 136  suits. 

Multiples.  Art.  153.  i— 144.  ^—374.  5—1,140.  4—5,200. 
5—10,230.      6—11,658. 

Art.  156.  i— 264.  ^—300.  5—48.  4—90.  5—189.  6—2,016. 
7—72.     5—3,450.     9—1,512.     i^— 3,600.    ii— 1,587,600.     i^— 19,008. 

Q  2 


370  ANSWERS    TO    WRITTEN  PROBLEMS. 

Art.  158.  i— 4,050.  ;e— 1,680.  5—1,340.  4—42,930.  5—2,520. 
e— 510.  7—2,520.  <5— 10,500.  9— $5.40.  i^— 960  bu.  ii— 240  min- 
utes ;  A,  20  miles ;  B,  15  miles  ;  C,  12  miles,      i^— 12,852  gal. 

Cancellation.  Art.  161.  i— 3.  2—b.  3 — 4.  4— 8.  5—9. 
6—851.  7—17.  8—2.  9—^\.  10—5.  11-7.  12—4:,  13—9, 
U—20.  i5— IJ.  16—Qj\.  17—1.  18— i.  19—S  lb.  20—22  cows, 
^i— 108bu.    ^^—96  mi.    23— QT.   ^4— $74.67.    ;^5— 40da.   fo— 30da. 

FRACTIOiSTS.  Reductions.  Art.  177.  i— J.  2—i.  3—i. 
4-1  ^-l  6-i.  7-1  8-jS.  9-j\.  10-^,  ll-\l  12-ii, 
13-j\.  U-\\.  i5-i|f .  16-i  yard.  i7— f  da.  i<^— $} ;  $t. 
19—1  lb.     20— P^.     21.  il  of  a  tou. 

Art.  179.  i-m-  2-\ii  3-^\.  4— /tV  ^-j?2-.  e-m. 

^-ill.  8—^'h%'   ^-IM.   ^^-A%.   ^i-rWj.   i^-C.   iJ-Crushed. 
Art.  183.    l-i^,  li,  f|.    ^--If,  i|.    ^-m,  «i.     -^-T^,  1%, 

a'tf,  iViT.  ^-ifii.  mi  HM,  jm.  e-Aw,  iWWy,  Am.  7- 

A«A,   ^VA,  A'b%,  AYt7,   ill*.      ^-AWA,  iim^,  /iWifir,  i^5V*, 
i%mh  Hii^'     9-i'^%  tm,  ^^,  ^ttfr.     i^.  23,400.     ii.  ,^^^, 

mn>  -ii^h^  ^\%>  mU' 

Art.  180.    i-ff,  H.    ^-H,  {^.    ^-iJ,  A,  M.    -^-A,  fi  if. 

^-ff,  i!>  i5.   e-it,  A>  H.   ^-A%,  iia  iijs,  im.   ^-"H* 

ff,  fi  ll>  if.     ^-f?.  ^,  ii  IS.     i^-T^W. 

Art.  188.  i— 176.  ^—18.  .5-266.  4— 18^.  5— llf|.  6— 233i|. 
7_58f.  S—4A.  9—lOii.  i^— 378.  ^i— 41^.  i-?— 62^^.  i^- 4,402. 
i^_276f.  i5— 180f  16—151.  i7— 168f.  18— So^.  19— Q25. 
S0—4SS^.  21—19i  T.  ^^— 586f  bushels.  ^,5— $488i.  ^4— $362f . 
^5— $945|. 

Art.  190.  1—^i^.  2—^.  3-^^.  4-Vi-.  ^— ^IF-  6-^, 
7_i|7.  8—^.  9—^^.  10— U%-  11—^'  ^^—HV-  i3—Hi^' 
■-'-        16—^l^K       17-iU-       ^S-HV'       19-H^' 


Addition.  Art.  193.  l—^\.  2—m  3—lj\%.  4— 7^.  5— 9f|. 
6-1^^.  7-26-m.  8-lU-  9-SiU-  ^^-246fi.  ll-4Sj\.  12-lS^. 
13-4^  yd.     i4-246f^  mi.     i5-|90iU' 

Art.  194.  1—m  A.  2-2^^  T.  3-2^^.  4-2^.  5-16||. 
e—2\^.  7— 3f .  ^—59^.  P— Entire  salary.  10—%2f^.  11— 5j^^  cd. 
-^^-3^^.  i^-liM.  U-^im-  i^-109ii.  i6- 140^  miles. 
Z7-137ff  yd. 

Subtraction.  Art.  190.  1—U-  ^—U-  ^—ts-  4— A-  ^— At- 
6-iWc.  7-^h'  ^-rh-  9-^.  10-iU%  miles.  ii-23f. 
12-62j\.  13-98^.  U-^m-  -^5-J.  ^6-18ifJ.  ^^-^A^- 
18-Q5mi'     19-297^i^. 


ANSWFES    TO    WRITTEN  PROBLEMS.  371 

Art.  197.  i-M.  -^-iff.  S-m^  4-mm'  ^-^SH.  6-26HI|. 
7-Tl,r  cd.  ^-MT.  9-3ff.  i^-7|||-.  11-^^^.  i^-$4^. 
i^-lSHfnii.  i^-35Myd.  i5-^^.  i^-^Vir.  ^7'-14f;-.  i5-6^. 
19-l^i^.  20-1^.  21— i^.  22-^%.  ^^-llSfi.  ^^-105,^. 
25—i^^K,     26-^\.     ^7-Iiicreased  i     ^5-3ff.     29-1^^;  ^;  1^ 

Multiplication.    Art.  199.    l—Ul     ^—tts-     ^—{i-     4—^M- 

12-1.  13-^.  U-^.  IS—i-^.  16-i^.  17-n.  18-1.  19-421 
20—11.  ^i— 348i  22— ii  of  a  melon.  23-^  of  the  factory. 
24—$l^~is-     ^^—U  A.     ^6—801  rd. 

Art.  200.  1—2^.  2—llj\.  S—l^.  4—^4.  5—370.  ^— S^V 
7— 238f.   5— 46,405f.   9—42j\h\i.   i^— $35|.   ii— 37fda.   12— $112^\. 

Art.  201.  i— 41  2— 90.  3—2^.  4—^lU'  5— $33|.  6— $25.81J. 
7—63,2791  ^_385,228if.  5— 262^  lb.  i^— 805f  lb.  11— $9.76. 
12—$1Q.  551      jr^_|386. 741.      i^— $4, 21 6. 40f . 

Art.  202.  i— |6f.  ^—$4^.  3—$j\.  4— $6^.  5— $8.33i. 
e— $7.38ff.  7— $56^.  8—ii-^.  9—U^  rd.  i^— $3ff.  ii— $71 
12-dQj%  mi.  IS-U.  U-^m-  ^^-iU'  ^^-tV  ^7-12.06}. 
i<?— $5.50.     i5— $lit^.     -^6^—225  yd. 

Division.    Art.  204.    1—t\%.    2—1-X.    S—tU-s-    4—h' 


6-im.  7-81  8-li.  9-12^\.  10-1^.  11-11  ^^-A.  ^3-1 
U—U-  ^'^— If-  i6—2^^\.  i7— 7f|.  18—42f.  19— Q3  sq.  yd. 
20—$jU'     ^^~80i  da. 

Art.  205.  1^%  2-^%.  3-j\.  4-Tih'  5—i^  6-^.  7-Hi. 
8-m.  9-11^.  i^-208ft.  ii-100^.  12-210^^%  IS—^-^T. 
U—i^  T.     i5— ft  lb.     IQ—h  lb.     i7— $1^^.     18—1691%  lb. 

Art.  206.  i— 15f.  ;^— 401  ^— 210f.  4— 73y\.  5— If.  6—371 
7—100.  8—2^.  9—%64l.  10—2^  h.  ii— 5J|  yd.  12—69^  doz. 
i^— 24  pairs,  i^— 35yVyd.  i5—32  blocks,  i^— 20|bu.  i7—5f  pages, 
i^— 2,112  steps.      i9— $22^.     20-$^^^. 

Art.  207.  1—ih'  ^-50f.  ^-405^  4-28.  5-^«^.  6-7,991if.' 
7—10.  8—^.  9—j%  yd.  i^— 3^-  A.  ii— 27i  quarts.  12—^  cd. 
i^— f|lb.  U—U^^'  1^—^lb.  16— Hi  da.  i7— if  lb.  18— 5^  da. 
i9— 35min.  -^^—$5%.  ^i— 15|-|  da.  22— $1.  ^^_214§  T.  24—^, 
25-^^.  26-m.  ^7-131  28-2i^.  29-^\.  30-^  Sl-^i^ 
32—1^. 

Review.     i-i|;  ^.    ;^-282ff;  450f f ;  182i|f.    ^"iff ,  HI,  ft*. 

•.^-99f|.     5— 82^V     6— 7,727i.     7—^.     8—%^^.     9— 8i|  pounds. 

i^— 123|i  acres,     ii— 91      i^— $10,5931.     13—%18-i^.     -^.4—1,360  lb. 

i5— $78.      i6— $16||.      i7— f ;  ^;  A;  f  *>  A-      -?^— Diminished  /^. 

19-i.     2o-ii^\%%.      21-1^;  A;  A.      ^^-3A;  iff;  m. 

^^_1^;_2_;||.    ^^-12f|;  2/^;  37^.    25-m'    ^^-^W>  A^,  ^^^ 


372  ANSWERS    TO    WRITTEN  PROBLEMS. 

;^7— 79||  yd.  28— i}.  S9—21^  acres  to  corn ;  21  f  acres  to  pota- 
toes  ;  7|  acres  to  wheat.  30— d^  boots.  31— Ul^.  32— B;  14^  mi. 
33—$27A0^.      34-m.      35— i^^. 

COMPOUND  NUMBERS.  Reduction.  Art.  220.  i— 353,91 9  in. 
;^— 985,140  sq.  in.  ^—1,053 J  feet.  >^— 22,873  sq.  ft.  5—8,831  qt. 
6—234,128  cu.  in.  7—1,539  in.  5—3,500,800  sq.  rd.  5—2,800  A. 
iO— 976  cu^ft. 

Art.  222.  i— 77  ft.  1  in.  S—Q  mi.  80  rd.  3—^  sq.  mi.  30  A. 
.4—33  sq.  yd.  5—72  rd.  2  yd.  6—540  bu.  3  qt.  7—191  gal.  3  qt.  1  pt. 
5—5  sq.  mi.  36  A.  5—3  cu.  yd.  26  cu.  ft.  624  cu.  in.  10—%  sq.  rd. 
16  sq.  yd.  4  sq.  ft.  32  sq.  in.  ii— 21  cd.  39  cu.  ft.  1,000  cubic  inches, 
i^— 119  bu.  2  pk.  1  qt.     i.^— 19  cd.  37  cu.  ft.     i^— 215  gal.  3  qt.  1  pt. 

Art.  223.  i— 6,408  tiles.  ;^— 508  pt.  ^—1,930  rd.  4— 48,400  hills. 
5—39  acres  146  square  rods.  6—9  bu.  2  pk.  1  qt.  7—320  bottles. 
5—507  pt.  papers.     5—91  gal.  1  qt.     i^— 5  bu.  2  pk.  1  pt. 

Addition.  Art.  226.  i— 208  gal.  3  qt.  <^— 39  A.  144  sq.  rd. 
5—55  cu.  yd.  16  cu.  ft.  1,007  cu.  in.  ^—14  yd.  6  in.  5—49  gal.  3  qt.  3  gi. 
6—81  bu.  1  pk.  4  quarts.  7—21  cd.  96  cu.  ft.  5—4  mi.  120  rd.  1  yd. 
5_2,041  bu.  3  pk.  2  qt.     10—^  gal.  2  qt.     ii— 115  sq.  yd.  5  sq.  ft. 

Subtraction.  Art.  228.  i— 2  miles  276  rods  2  feet  9  inches. 
2 — 5  sq.  rd.  29  sq.  yd.  2  sq.  ft.  36  sq.  in.  5—5  yards  1  foot  5  inches. 
^_4  bu.  2  pk.  3  qt.  5—50  cu.  yd.  8  cu.  ft.  762  cu.  in.  6—87  cords. 
7—1  ml  174  rd.  3  yd.  2  ft.  6  in.     5—14  gal.  1  qt.  1  pt. 

Multiplication.  Art.  230.  i— 853  bu.  6  qt.  ;^— 1,098  mi.  3  rd. 
3  yd.  2  ft.  6  in.  5—1  cu.  yd.  1  cu.  ft.  1,152  cu.  in.  ^—292  A.  80  sq, 
rd.  14  sq.  yd.  2  sq.  ft.  32  sq.  in.     5—230  gal.  1  pt.     6—848  mi.  226  rd. 

3  yd.  7—24  cd.  24  cu.  ft.  5—37  bu.  3  pk.  4  qt.  5—1  A.  60  sq.  rd. 

4  sq.  yd.  4  sq.  ft.  72  sq.  in.  iO— 589  mi.  103  rd.  2  yd.  2  ft.  6  in. 

Division.  Art.  232.  1—1  mi.  259  rd.  2  yd.  2  ft.  lOf  in.  2—2  sq. 
mi.  365  A.  119  sq.  rd.  17  sq.  yd.  2  sq.  ft.  82f  sq.  in.  5—3  cu.  yd.  3 
cu.  ft.  288  cu.  in.  ^—43  rd.  4  yd.  2  ft.  If  in.  5—21  A.  m  sq.  rd.  136 
sq.  ft.  46|-  sq.  in.  6—1,340^^5  cu.  in.  7—6  kegs.  5—2.  5—2  rd.  2 
-yd.  2  ft.  10—12  cu.  yd.  12  cu.  ft.  ii— 65  bags,  i^— 60  sq.  rd. 

Art.  234.  i— 9,683  lb.  <^— 412  oz.  5—34  bar.  147  lb.  ^—20  bar. 
63  lb.  7  oz.  5—58  T.  10  cwt.  3  lb.  9  oz.  6—1,785  lb.  12  oz.  7—60  T. 
472  lb.  8  oz.  5—281  lb.  3}  oz.  5—47  lb.  2  oz.  i^— 41,716  packages, 
ii— 26  T.  812  lb.  8  oz.  12—^  T.  8  cwt.  46  lb.  i5— 36  lb.  12  oz. 
i^_53  T.  875  lb.  i5— 1  T.  1,825  lb.  i6— 360  da.  i7— 33  lb.  14  oz. 

Art.  235.  1—1  wk.  3  da.  3  h.  14  min.  35  sec.  ^—5  wk.  5—8,784  h. 
^—31,556,929  sec.  5—1,047  da.  17  h.  54  min.  6—7  wk.  5  da.  3  h.  4 
min.  51  sec.  7—12  da.  6  h.  40  min.  5—329  yr.  119  da.  22  h.  12  min. 
5—267,120  min.  ^6*— 11  da.  13  h.  46  min.  40  sec.  ii— 5  jr.  4  da. 
i^— 39  h.  20  min.  i5— 8  da.  1  h.  15  min.  i4— 4  da.  7  h.  30  min. 

Art.  236.  i— 84  yr.  3  mo.  ^— Feb.  22, 1732.  5—1  yr.  9  mo.  27  da. 
^-—4  mo.  24  da.  6—4  yr.  1  mo.  14  da. 


ANSWBES   TO    WRITTEN  PROBLEMS.  373 

Art.  237.  i— 498  doz.  ^—3,739  sheets.  3—2  grt.  gro.  2  gro.  10 
doz.  8.  4 — 11  rm.  1  quire  15  sheets.  5 — 25  rm.  18  quires  16  sheets. 
6 — 15  grt.  gro.  1  gro.  4  doz.  7 — 480  sheets.  8 — 8,540  clothes-pins. 
9 — 10|-  doz.  pairs.     10 — 5  gro. 

Review,  i— 31  bu.  1  pk.  2  qt.  1  pt.  ^—2  gallons,  .f— 250  days. 
4— lOgal.  Ipt.  5— 84bu.  8pk.  4qt.  5— 15,0001b.  7— 24  rd.  1  yd.  1  ft. 
^—1  A.  97  sq.  rd.  218  sq.  ft.  86  sq.  in.  5— $245.  10—2  cd.  96  cu.  ft. 
ii— 59  bar.  75  lb.  1^—Q  quires  12  sheets.  ^^—285  bu.  i^— 1,277 
cu.  yd.  21  cu.  ft.  i5— 81  mi.  120  rd.  i6— 86  A.  187.25  sq.  rd.  i7— 12  da. 
i<^— 204  T.  1,050  lb. 

MEASUREMENTS.  Rectangles,  Triangles,  and  Trapezoids. 
Art.  247.  i— 824  sq.  rd.  ^—80,100  sq.  yd.  ^—2,175.5  sq.  ft. 
4—808.052  sq.  in.  5—11  sq.  ft.  40|  sq.  in.  6—8,904.5  square  rods. 
7—1^449.6  sq.  yd.  8—7  sq.  ft.  P— 80,174  sq.  ft.  i6'— 1,389.625  sq.  mi. 
11—21  sq.  rd.  26  sq.  yd.  18  sq.  ft.  72  sq.  in.    i^— 6,400  sq.  ft.    i^— 800  A. 

Art.  248.  i— 86  rods.  ^—22  rods.  <?— 49if  rods.  4— 30^^  rods. 
5— 1,192  ft.  6— 822^2^  ft.  7— 162.4125  ft.  5— 112^^6  ft.  (=112.0068 +ft.). 
9_304if4  rd.  it*- 176.65+rd.  ii— 565. 7456+ rd.  i^— 327. 8898+ rd. 
13— S^  feet.  U—m%  yards.  15—8^  ft.  16— Sj^-^  ft.  i7— 9f f  ft. 
18—12^  ft.     19— W  ft.     ^^—27  ft. 

Art.  250.  i— 1,134  sq.  in.  ;^— 4,608  sq.  ft.  ^—2241^^  sq.  ft. 
4— l,185|sq.ft.  5— 8,948sq.  rd.  6— 14f  in.  7— 4  ft.  11  in.  5— 42  ft. 
9—24  ft.      10— m^  rd. 

Art.  251.  i— 270  sq.  in.  ^—575  square  feet.  .?— 85,805  sq.  yd. 
4—148  sq.  ft.  88  sq.  in.     5— 128i  sq.  ft.     6—509.90625  sq.  ft. 

Art.  252.  i— 2401  sq.  in.  ^—775  sq.  yd.  .^— 80.164yV  sq.  mi. 
4— 100|§  sq.  rd.     5— 46J  sq.  yd.     6— 91f  sq.  ft. 

Art.  253.  i— 203  A.  145  sq.  rd.  ^— 81|fyd.  5— 18  ft.  9  in.  by  40  ft. 
^_253i  sq.  yd.  5—351  feet.  6—49  sq.  ft.  94  sq.  in.  7—41.4  feet. 
^—$5,862.50.     5— 18f  I  rd.     i^— 2,495  ft. 

The  Circle.  Art.  255.  i— 25ift.  ;^— 8ft.4fin.  ^— 53  ft.  5^  in. 
^_3  ft.  4f  in.  5—9  ft.  6^  in.  6—5  ft.  3  in.  7—170  yd.  4.9  +  in. 
8-5^  in.       * 

Art.  256.  i— 10,028|  sq.  ft.  ;^— 201i  sq.  in.  J— 707A  sq.  in. 
.^— 41,367|+sq.  rd.     5— 8, 148. 728+ sq.  rd.     6—79,577.45  sq.  yd. 

Art.  257.  i— 16.9964  +  mi.  ^—170.77  +  sq.  ft.  3— 2544  sq.  ft. 
.4— 753.758  +  sq.  rd.     5— 201^  sq.  in. 

Rectangular  Solids.  Art.  260.  i— 1,155  cu.  ft.  ^—3,052.5  cu.  ft. 
,^—861  cu.  ft.  4—2,002  cu.  ft.  5—8,944.2  cu.  ft.  6—5,184  cu.  in. 
7_1^000.82  cu.  in.  5—1,377  cubic  inches.  ^—1,252.8  cubic  inches. 
^^_64.21825  cubic  mches.  ii— 282,000  cu.  yd.  i;?— 87,000  cu.  yd. 
i^— 889,648  cu.  yd.     i^— 860,000  cu.  yd.      15—197,640  cu.  yd. 

Art.  261.  i— 64  inches.  ^— 45f  in.  3—U^\  in.  .4— 86.4  in. 
S—42iiim.     6— 125  ft.     7— 187i  ft.    .<5— 78.48137  +  ft.     9— 11 2^  ft. 


374:         ANSWERS    TO    WRITTEN  PROBLEMS. 

^^_105f5-  feet.     11— n\  inches.     12— \%  ft.  3?  in.      13—4:  ft.  83^  ia 
^^_2|8  in.     i5— 92|  ft.     16— ^  ft. 

Art.  262.  i— 84  loads.  ^— 9||  cd.  .^—1,728  cubes.  ^— 137JI-J 
cu.  yd.     5—36  ft. 

Business  Measurements.  Art.  265.  i— 21,489.3  cubic  inches. 
2—1  gal.  1.2367  qt.  5— l,289yV  bu.  ^—2,240  sq.  ft.;  22.40  squares. 
5 — 33  squares  ;  6,187^  square  feet.  6 — 3,456  slates.  7 — 67^  bunches. 
^_|76.80.  i?— 23.6  bunches,  i^— 677^  square  yards,  ii— 1233*. 
X^_460 1  cubic  yards,  i^— $5,898.75.  i^— 50  panes.  iJ—12f  boxes. 
16—%^M\.  i7— $07.55.  i«9— 431^}  cubic  yards.  iP— 444^  cu.  yd. 
^^_13,48ii,8  cu.  yd.  ;^i— 27  ft.  22—12  hhd.  i*.^— 256.4749  +  bar. 
;^4_6'7^\2.  square  feet.  i*5— 14,580  bricks.  f6— 113|  cubic  feet. 
;^7_319i  perches;  2S^i^  cd.  ;t^<^— $16.18^  ;^i?— 45  feet;  540  feet. 
^^_3,648  bricks.  Ji— $559.10.  ,?^— 3,226}  cu.  yd.  J5— $17.82. 
^4_38-4  perches.  ^5—160  sq.  ft.  36— ^Q  sq  ft.  37— UH  cu.  ft. 
^*,^_400  sq.  ft.  J5— 180  sq.ft.  4(>— 112  ft.  ^i— $30.37^.  4^— 501.0  bu. 
^_l,024bu.  44— 280bu.  .^—6,528  bu.  .^C— 502.2  bu.  .^7— 66|  T. 
^-l^T.     45-2f|T. 

Revieio.  i— 1,512  square  rods,  j^— 918f  sq.  rd.  .^—2,714.4  sq.  rd. 
^_1, 649.375  sq.  rd.  5— 35|  sq.  yd.  6— 31f  sq.  yd.  7— 42^^^  sq.  yd. 
^__37isq.yd.  9— 209.035  sq.  yd.  i^— 82.80 +sq.  yd.  ii— 247.35  sq.  yd. 
i^— 97.9801  +  sq.  yd.  i^— 31icu.ft.  i^— 51fV  cu.  ft.  i5— 51y»5  cu.  ft. 
i6— 85/4  cu.  ft.  i7— 140^5^5^  cu.  ft.  i^— 290^^^^  cu.  ft.  i5— 171|  cu.  ft. 
;^^_255.^4^  cu.  ft.  ^i— 15i^V  cu.  ft.  22— in^  cu.  ft.  ;^^— 265  sq.  in. 
^^—13.3656  + A.  1^5—36  sq.  in.  1*^—2,150  persons.  ^7— $49.87^. 
S8—%m.'^().  29— IQ  ft.  .^^—8^-  yards.  31—9  in.  32—10  ft.  8  in. 
^^— OOOrd.  ^^— 18ft.  .^5— 6 ft. 4 in.  .?(?— 2,046 pickets.  ^7— 288 sq.ft. 
^5_o4  A.  108  sq.  rd.  39— 24  it.  ^— 13  sq.  yd.  4^— 4  sq.  ft.  24  sq.  in. 
^— 160rd.  .^<9— 32y\sq.rd.  44— $67.20.  45— 310sq.rd.  46— 468  cu.  ft. 
47_4  cu.  ft.  108  cu.  in.  4^— 12|  hours.  49—2  ft.  6  in.  5^— 8^^  ^eet. 
Sl-Sdl^hu.  5^— 163^^  sheets.  5^— $32.50.  54— $.21.  55— $23.94. 
S6— 427m  bar.  57— 6it  in.  5.9—134,744  bricks.  S9—^  of  the  wall. 
60— $140 M.  61— $76.82.  62—$2M.  6^— 55^  T.  (?4— 3,360  ft. 
65— $1.36|.  66— $18.62.  67— 56^  bushels ;  45^^  bushels.  6<?— $81.04. 
69— $12.80.  7^— $466.36.  7i— 48t^\  cd.  7^— 33i^  T.'  7c?— 576  bu. 
75—10  ft.  3i  in.     76— 20^  in.  by  30^  in.  by  42|  in. 

PERCENTAGE.  The  Five  General  Cases.  Art.  276* 
^_38  sheep.  ^—110  pupils.  <?— 3.3  s^al.  4— 14,400  A.  5— l,234f  bu. 
g_7,350nien.  7— 2,062iLehigh;  2,875  Lackawanna;  l,312iPittston. 
^— 789|  bu. 

Art.  278.  i— 6^.  2—^%.  3—%%%.  4—1500^.  5—20^.  6—35,^. 
7—8^.     8—18%.     9—7Ufc.     10—Q^fc. 

Art.  280.  i— 1,300  tons.  ^—92  miles.  ^— $2,777|.  4— $1,900. 
5—2,000  sacks.  6— 111^.  7—125  pupils.  .9—125  sq.  rd.  P— $.25. 
10—3^  lb. 

Art.  282.    i— $8.40.    ;^— $492.    ,^— 82i^.    4— 1328.50.    5— $266| 


ANSWUnS    TO    WRITTEN  PROBLEMS.  375 

Art.  284.  i— $450.  ^—$337.50.  ^—$1,875.  ^—$1,680.  5— $1,750. 
e— $90.     7—42.9  yd. 

Special  Applications.  Art.  287.  i— $.25.  ^—$.58.  5— $.09. 
4— $340.  5— $23.43f.  6—^\<p.  7—650.  <?— 80^.  5—274.4  lb. 
i6/— $.663.  ii— $2.08.  i^— $3.25.  i^— $2.  i^— 850.  15— ^%. 
16—\Q%%.  17—11^1^%.  18—^^fc.  19—25fc.  W—17m.  ^l—20fo. 
^^—$95. 62}.     f ^^— $208. 33}.     ^^— $  .  64f . 

Art.  290.  i— $20.75.  ^—$209.06}.  5— $45.50.  ^—$59.25. 
5— $75.37}.  e— $19.40|.  7— 12i^.  8—2^ff^fr.  P— Investment, 
$1,323.81;  Com.,  $66.19.  i^— Investment,  $12,000;  Com.,  $400. 
ii— $2,565.85.    i-^— $5,490.20.    i«?— 20,0001b.    i^— $278.    i5— $6,004. 

i6— $2,475. 

Art.  294.  i— $90.  -^— $39.29|.  5— $29.25.  ^—$9.37}.  5— $84.37}. 
e— $45.  7— $731.25.  8—ifc.  9— life.  10—1%.  11—2%.  i^— $1,266. 66f. 
i^— $4,500.     i^— $1,950. 

Art.  295.  ^—$37.37}.  ^—$3,728.  S-l-fi^i.  4—i%-  5—1,%. 
6— A,  $4.92;  B,  $5.91;  C,  $8.86;  D,  $2.95;  E,  $16.73.  7— $1,425.23. 
^—$9,350.      9—2%;   $44,  $6,  $16,  $30,  $10,  $24,  $8,  $12. 

Art.  303.    i— $140.62}.    ^—$218.04.    ^—$963.83.    .^—$2,169.07. 

Art.  311.  i— $107f  per  share.  ^—$84  per  share.  ^—$35.87}. 
^— $107.18f.  5— $8,625.  6—87  shares.  7—25  shares.  8—6U%. 
9—6-1%.     i^— $3,393.75.     ii— $1,741.87}.     i-^— 15  shares. 

Art.  318.  i— Keed,  $1,008.82;  Clark,  $1,441.18.'  ^— A,  $6.25; 
B,  $8.75;  C,  $7.50.  ^—Hanover,  $4,950;  Globe,  $3,960;  ^tna, 
$3,300;  Franklin,  $2,970.  ^— Bates,  $354.86;  Davis,  $473.14. 
5— Capital:  G's,  $20,000;  H's,  $30,000;  K's,  $25,000.  Dividends: 
G's,  $4,960;  H's,  $7,440;  K's,  $6,200.  6— E,  672  bales;  F,  400  bales; 
G,  552  bales;  H,  296  bales.  7— W.,  $11,700;  J.,  $26,325;  B.,  $32,175. 
^— A,  70  bii. ;  B,  87}  bu.  9— A,  24if  shares  ;  B,  47^^  shares  ;  C, 
28xV  shares,  i^*— Hilton,  $23.58;  Roberts,  $26.42.  11— A,  $2,982^; 
B,  $2,455f ;  C,  $2,2803^;  ^>  I^Olf.  12— A,  $647.92;  B,  $161.98;  C, 
$323.96;  D,  $809.90. 

Interest.  Art.  325.  i— $28.02.  ^—$16.05;  $337.05.  ^—$58.62}. 
^—$147.06.     5— $182.47.     6— $4,312.50.     7— $2,655.     5— $78.41. 

Art.  327.  i— $14.28.  ^—$3.13.  ^—$54.53.  ^—$272.09. 
5— $301.61.      6— $3,523.35.      7— $2,759.32.     ^—$294.     P— $60.55. 

Art.  329.  i— $71.82.  ^—$45.84.  .?— $303.94.  4— $44.76. 
5— $3.64.      6— $35.27. 


Art.   331.      i— $193.75. 

;^— $106.56.       5— $63.61. 

^—$36.92. 

5— $119.26.        6— $135.63. 

7— $74.59.        <9— $44.53. 

9— $25.84. 

i^— $83.48.       ii— $87.19. 

i^— $47.95.       IS— $2S.6S. 

i^— $16.61. 

i5— $53.67.      i6— $290.63. 

i7— $159.84.      i<§— $95.42. 

i^— $55.38. 

^^—$178.90.      ^i— $116.25. 

^^—$63.94.     -^^—$38.17. 

^4— $22.15. 

^5— $71 .  56.      ^6— $146. 50. 

^7— $80.58.      ^<?— $48.10. 

-^5— $27.92. 

876  ANSWERS    TO    WRITTEN  PROBLEMS. 

5^_|90.18.  ^i— $102.55.  5;?— $56.40.  ^<?— $33.67.  34—%\^M. 
^5— $63.13.  J6— $65.93.  57— $36.26.  5.9— $21.65.  <?5— $12.56. 
.4^— $40.58.  ^i— $219.75.  .^—$120.86.  .f?— $72.15.  ^^—$41.87. 
.^—$135.27.  >^(5— $87.90.  >^7— $48.35.  .4^— $28.86.  ^—$16.75. 
50— $54.11.  57— $3,730.57.  5;^— $2,051.81.  55— $1,224.87.  5^— $710.88, 
55— $2,296.37.         56— $2,611.40.  57— $1,436.27.         55— $857.41. 

55_|497.62.  6?(?— $1,607.46.  6^i— $1,678.75.  6^— $923.31.  65— $551.19. 
6^— $319.90.  6*5— $1033.37.  66— $5,595.85.  67— $3,077.72 

65— $1,837.30.  69— $1,066.32.  70— $3,444.56.  7i— $2,238.34. 
7^— $1,231.09.  75— $734.92.  7.^— $426.53.  75— $1,377.82. 

76— $6,004.54.  77— $3,302.50.  75— $1,971.49.  75— $1,144.20. 
56— $3, 696. 1 3.  5i— $4, 203. 18.  5-^— $2, 31 1 .  75.  55— $1 ,  380. 04. 
54— $800.94.  55— $2,587.29.         56— $2,702.04.         57— $1,486.12. 

55— $887.17.  55>— $514.89.  i>6— $1,663.26.  5i— $9,006.81. 

92— %4:,  953. 75.  55— $2, 957,24.  5^— $1 ,  71 6. 30.  55— $5, 544. 19. 
56— $3,602.72.  57— $1,981.50.  55— $1,182.89.  55— $686.52. 
100— $2, 21 7. 68.  101— %2, 365. 94.  102— %\,  301 .  26.  755- $776. 82. 
i^^_|4r,0. 84.  i55— $1 ,  456. 36.  i66— $1 ,  656. 1 5.  i67— $910. 88. 
^^^—1543. 77.  765- $315.59.  776- $1,019.46.  iii- $1,064.67. 
n2—%T^^n.  57.  ii5— $349. 57.  ii^— $202. 88.  ii5— $655. 36. 
ii6— $3, 548. 90.  ii7— $1 ,  951 .  90.  ii5— $1 ,165. 22.  ii5— $676. 26. 
i;?0— $2,184.55.  i;^i— $1,419.56.  i^^— $780.76.  7^*5- $466.09. 
i^.^— $270.51.  it'5— $873.82.  ii'6— $85.08.  i^7— $95.06.  i^5— $88.17. 
ij^5— $85.97.  155- $99.27.  i5i— $90.09.  i5^— $85.46.  i55— $96.87. 
i5^— $88.99.  i55— $327.07.  756— $365.43.  757— $338.94.  755— $330.46. 
755— $381.60.  7^6>— $346.29.  7^7- $328.52.  7^— $372.36.  7^5- $342.09. 
744— $2.12.  745— $2.37.  746— $2.20.  7^7- $2.14.  7^— $2.47. 
i^_|2.24.  755— $2.13.  757— $2.41.  75^— $2.22.  755— $654. 
75^— $204.  755— $22.  756— $131.25.  757— $145.60.  755— $1,957.04. 
755— $219.15.  765— $5.94.  76;^— $39.20.  76^- $1,090.  765— $810.55. 
766— $1 ,  388. 16.     767— $1 ,  164. 46.     765— $2, 970. 18.     765— $669. 30. 

Review.  7—21,525  votes.  ^— 1st,  36^;  2cl,  34^;  3d,  30^.  5—36  mi. 
^_|.333^  5— Loss,  $2.  6—12^%.  7—$  .90.  5— $6,310.  5~$118.75, 
fees;  $831.25,  remitted.  75— $44,314.29.  77— $254.94.  7;^— $120.75. 
13—21^.  7^— $2,287.50.  75— $590.  76— $6.83.  77— $99.96. 
75— $436.32.  75— $2,175.  20— $1,200.  21— $2.1Sl.  22—12%. 
23—2(1^  apiece  ;  30^  a  dozen.  ^4— $9,800  ;  $6,500  ;  $10,250  ;  $7,150. 
;^5— 922.722  +  bushels.  ^6— $2,650.  -^7— $2,250.  ^5— $5,941.25. 
;^9_|841.42.  57—^  of  the  ship.  5^—200  bushels.  55— 14f^. 
^^_|6,958.40.  55— li^.  56— $1,004.25.  57— $675.  55—230%. 
55—10^.  45— $282.49.  ^7—50^.  42— \X  is  $92.60  per  amium 
cheaper  for  him  to  buy  the  farm. 


ANSWERS   TO    WRITTEN  PROBLEMS.  377 


SUPPLEMENT. 

METHODS  OF  PKOOF.  Art.  5.  i— 354,455.  ^—5,084.4751. 
S—m-.  ^—$18,330.63.  5—227,933.  6—75.7917.  7—iii^.  ^—3,121,788. 
9—167.18091.  i{?— $4,426.98f.  ii— fffff.  i^— 68.  i^— 69^^. 
i^- 5.242  +  .     i5— 6.42  +  .     i6'— 36.4.     17—4:11 

SHORT    METHODS.      Art.    12.       i— 308,520.        ^—4,557,168. 

.^-202,875,372.      4— 18,438.      5—86,922.      6—1,282.68.      7—2,800.28. 
5— .00030848. 

Art.  10.  i— 1,940.  ^—43,375.  ,^—245,500.  4—455.  5—5,037.5. 
6—119.     7— $40.62i.    5— $6,250.    9— $660.    i^~$1.80.    ^i— $468.75. 

i^— $423. 

Art.  17.  i— 742,104.  ^—6,835,158.  .^—169,693,029.  4—8,020,619,793. 
5—227, 640, 772, 359.     6—2, 694, 251, 157, 219. 

Art.  18.  i— 725.  ^—66.  .^— 3,737|.  4-5,005f.  5—10.24. 
6-80.5.      7— 34iV 

Art.  19.  i— 1,712.  ;^— 26^^.  .f— 198.  4—7.14.  5— $38.28. 
6— $1,774.  7— $2.27.  ^—$.1127.  5— $65.  i^— 3.4  yd.  ii— 65  lb.  15  oz. 
i^— 578.46  bar. 

CONVERSE  REDUCTIONS.  Art.  21.  i— f.  ^-|f.  3—U. 
^-ii    5-^^.    6-3^0-    7-m.    8-^i^rm.    5— AV  wk.    10-% 

Art.  22.  i— .35.  ;^— .40625.  .^-.2727+.  4— -0004.  5— 18.75;  7.075. 
6— .555  +  .     7— 19.3636+. 

Art.  23.  1—19  hours  12  minutes.  2—1  foot  10^  inches.  .^—70  rd. 
4—21  bu.  3  pk.  4  qt.  5—2  qt.  1  pt.  IJ  gi.  6—2.5344  square  inches. 
7_273  da.  18  h.     ^—13  cwt.  1  qr.  9-i-  lb. 

Art.  24.  -?— .921875  bu.  <^— .90625  gal.  <^— .625  gro.  4— .8282+rd. 
5— .6625  ream.     6— .10175  T.     7— .66  da.     <?— .5  rd. 

Art.  25.  i— 6doz.4f.  ^— 426  A.  106  sq.  rd.  20  sq.  yd.  1  sq.  f t.  72  sq.  in. 
5—146  sq.  rd.  20  sq.  yd.  1  sq.  ft.  72  sq.  in.  4— 1  bu.  2  pk.  1  pt. 

5—60  gal.  2  gi.      6—219  da.  14  h.  24  min. 

Art.  26.  1-M  gal.  ^-^rV  sq.  rd.  3-^^^  mile.  4— f|  bu. 
6-m  cu.  yd.     6-f  If t  da. 

PRICE,  QUANTITY,  AND  COST.      Art.  32.      i— $3,029.81}. 

^—$10,336.50.     .?— $1,531      4— $70.76i.     5— $167,471     6~$294.18|. 
7- $640. 74.     .9— $18,096.     S>— $64.35. 

Art.  33.  i— 429  bar.  ^—183.76  T.  S—4i  yd.  4—3,057  pickets. 
5— 385  1b.    6— 85,432  bricks.    7— 15,690  ft.    ^—2,784  lb.    5— 4,680  lb. 

Art.  34.  i— $7.06}.  ^—$56.25.  .^—$4.50.  4— $2.25.  5— $11.50. 
€—$7.50.     7— $18.75.     ^—$13.     5— $27.50. 


378  ANSWERS   TO    WRITTEN  PROBLEMS, 

Art.  35.  4—$SA9l  5—26  gallons  1  quart.  ^—$87,425.  7— |  cd. 
5— 111.61|.  5— $2.25.  10—^1.85.  ii— $41.58.  i^— 100  M  feet, 
i^— $125.  i^— $55.78^.  iJ— lO^Mft.  i6'— 3  T.  1,875  lb.  i7— $158.77. 
18— Q2^\h.  19—$/S2}.  ^0— $262.60.  ^i— 5,750  feet.  ;^^— $72. 
;^^— $902.     ^^—$32.85.    ;^5— 13  cwt.  1  qr.  21^%  1^. 

LONGITUDE  AND  TIME.  Art.  66.  i-8°.  ;^— 13'30".  5—1°  20'. 
^—7°  11' 15".  5— 17°  7' 30".  6— 123°  6' 45".  7— 3°20'W.  ^— 17°30'E. 
5—100^  3'  W.     i^— 30°  E.     11—6°  13'  W.     i^— 137°  17'  15  "  E. 

Art.  67*  1 — 5  hours  17  minutes  21  seconds.  2—6  h.  50  min.  28  sec. 
5— 5  h.  59  min.  47yV  sec.  ^— 10  h.  26  min.  3  sec.  .5— 6  h.  32  min.  12|  sec. 
6—6  h.  8  min.  1  sec.  7—11  min.  58  sec.  P.M.  8—8  h.  46  min.  25  sec.  A.M. 
9 — 1  hour  6  min.  17  sec.  P.M.  10—9  hours  40  min.  44  sec.  A.M. 
11 — 7  hours  41  min.  58  sec.  A.M.  12 — 4  hours  16  min.  25  sec.  A.M. 
13—8  h.  36  min.  17  sec.  A.M.     U—6  h.  10  min.  44  sec.  A.M. 

INTEREST.  Art.  87.  1-^%.  2-4:\%.  S-^%.  4-2t%.  5-9%. 
6—18i%.  7— 12|f^.  ^— $5.37i.  9— $84.37J.  i^— $60.83.  ii— $44.21. 
i<^— $41.14.  iJ— $645.81.  i^— $465.62.  i5— $338.37.  i6— $13.53. 
i7— $212.41.  i<5— $153.15.  i^?— $111.29.  ;^^— $259.68.  ^i— $4,076.33. 
;^^— $2, 938. 98.     23— $2, 135. 79. 

Art.  88.  i— $2.51.  ^—$234.  5— $15.78.  4— $.32.  5— $5.  6— $1.50. 
7— $330.38.  5— $8.33.  5— $2.84.  76^— $15.47.  ii— $13.23.  i^— $111.72. 
i.?— $5.26.     i-^— $199.60.     i5— $40.82.     16— $66.12. 

Art.  92.  i— $406.  ;^— $7,899.85.  5— $10,000.  ^—$257.14. 
5— $2,204.23.     6— $36.50. 

Art.  94.  i— $483.62.  ;e— $329.96.  ,^—$621.21.  .4— $67.50. 
5— $1,592.     6— $419.84. 

Art.  95.    l—4i%.     2—6%.     3—6%.    4—8.3^.     5-9%.     6—Z\%. 
Art.  96.   1—1  yr.  9  mo.  8  da.    2—2  yr.  6  mo.    3—22  yr.  2  mo.  20  da. 
^—8  yr.  4  mo.     5—6  mo.     6—10  mo.  15  da. 


PllOBLEMS   IN    THE    SiX  GE1^^ERAL   CaSES    OF  INTEREST.       1- 

;^— $63.73.  ^—$90,075.  4— $131.25.  5— $1,264.64.  6— $1,500. 
7— $666.67.  ,?— 5yr.4mo.  S>— 4yr.  2mo.  10—8fr.  11—4%.  i^— $400. 
i^— $810.  i4— $982.43.  i5— $1,413.45.  i6— $8,930.85.  i7— 1  yr.  6 
mo.  6  da.     i<?— I  rec'd  2^% ;  Buyer  of  note  rec'd  12^^-^. 

Exact  Interest.  Art.  98.  i-44.15.  ^—$8.97.  .?— $314.90. 
4— $253.37.  5— $1,414.83.  ^—$8,568.36.  7— $489.65.  5— $28.51. 
S>— $2.47.     i^— $37.26.     i^— $166.86. 

DISCOUNT.  Art.  109.  i— $6.75.  ;^— $62.50.  5-$28.03.  .4— $4.67, 
5— $1.75.  6— $2.67.  7— $10.26.  ^-$26.68.  9— $25.56  ;  $826.44. 
i^— $38.91;  $938.92.    ii— $20,621;  $1,479. 87i.    i^— $78.50;  $2,376.50. 

Art.  118.    i— $6.46.     ^—$3.82.    5— $4.92.     4— $24.94.     5- 
6— $596.70.  7— $814.87.  5— $8,428.25.   9— $416.99.  it^— $7.75;  $492.25. 
ii— $7.44;  $842.86.     i^— $239.17;  $9,760.83.     i.?— $17.50;  $1,232.50. 


ANSWERS   TO    WRITTEN  PROBLEMS.  379 

Art.  121.  i— $1.04.   .^—$50.87.   ^-$6.64.    .^—$17.05.   5— $908.75. 

6— $3,346.08.         7—11,073.68.         5— $87.58.         S>— $84.47;    $840.53. 
i6'— $77.61;  $2,914.93.     ii— $100.17;  $499.83.     i^-$53.48;  $506.52. 

Problems  m  Discount,  i— $909.72.  ^—$59.87.  «?— $4,345.38. 
.4— $2,345.38.  5— $1,615.95.  6— $135.53.  7— $2,740.17.  <5— $401.79. 
10— ^o  differenoe. 

COMPOUND  INTEKEST.    Art.  122.    ^-$743. 73.     ;^— $722.88. 

.?— $32.83.        4— $30.32.       5— $927.17.        6— $422.91.        7— $314.39. 
^—$146.58.     5— $63.22.     iC— $33.64.     ii— $72.92.     i^— $108.24. 
Art.  123.  '  i— $280.81. 

PAKTIAL  PAYMENTS.  Art.  125.  i— $1,705.54.  ;^— $1,218.15. 
.^—$283.75.     4— $11,990.22.     5— $542.12.     6— $1,643.92. 

Art.  126.    i— $678,121     ^—$1,898.70.     ^—$2,128.12. 

Art.  129.  i— $268.80.  ^—$66.03.  5— $58.08.  4— $35.96. 
5— $141.53.  6— $158.92.  7— $22.72.  5— $357.61.  5— $769.98. 
i^— $2,333.07. 

BONDS.  Art.  138.  i— $337.50.  ;^— $118.25.  5—236  bonds. 
4—34  bonds.  5—240  bonds.  6—64  bonds.  7— Mich.  7's,  fff^. 
8— Geo.  7's,  fll^.  9— 5's,  ^^%.  10— 4^%  ^%%.  ii— $3,247.50. 
i^— $21,200.     i^— 5f^.     U—^^N^fc. 

EQUATION  OF  PAYMENTS.  Art,  14:0.  1—4  mo.  ^—3  mo. 
29%a.  after  to-day.     <?— May  J. 

EXCHANGE.  Art.  149.  i— $251.25.  ^—$467.91.  5— $1,523.75. 
^_|615.23.  5— $330.65.  6— $2,708.75.  7— $115.  <?— $998.75. 
9_|839.82.    i^— $3,606.66.    ii— $506.    i^— $1,743.68.    ^5— $1,996.34. 

i4_|l,747.38.  i5— $543.97.  i6— $2,444.72.  i7— $723.79. 

i<^_|2, 440.20.      i5— $5,902.07.      ;^^— $1,365.59. 

RATIO  AND  PROPORTION.  Art.  155.  i— 3.  2—\.  S-^. 
4— f.  5—1.  6—1.  7— f  8—S.  9—i.  10—3.  11— $400.  if— 20  rd. 
i^— 72.    i^— 16  to  24.    i5— 4  ft.  to  3  yd.    16—^^%    17—90.    18-    " 

Art.  161.     1—35.     f— 12.     3—9.     J^—\^.     5—36.     6—15. 
5—32.      9—4%.      10— ^.      11—49.      if— 26.5.      i«?— 10.8. 
i5— If.     16—1  qt.     i7— 4  lb.     i^"- 5J  oz.     19—1.     20— 3\. 

Art.  164.  i— $8.  f— $1.33.  5— $56.37i.  4— $9.80.  5— $34.12^. 
6— $56.25.  7— $13.42.  5— $27.62^.  5— $2.22.  iC*— 6, 298xV  revolutions, 
ii— l,465fO-yd.  if— l,225imi.  i^— $176.32.  i^— 95.2  ft.  i5— 66|lb. 
i6— 39|qt.  i7— $160.71.  i<^— $993i  i5— $1.40.  f  ^— 20. 309  +  cd. 
fi— $918.  22— \  year,  f^— $22.  f^— $18.56^.  f5— 559||  miles. 
26—\3\  da.     f7— 427  A.  122  sq.  yd.     f<?— $61.38. 

Art.  166.    i— 18.     ^— 22i.     3—^oz.    4—2^-     5—^.     6—175. 

Art.  168.  i— $13.  f— 8,925  pumps.  5— $787.12.  4— 57-H  days. 
5— $120.     6—434  boards.     7—60  pairs.     5— $16.61.     S>— 8,505  feet 


380  ANSWERS   TO    WRITTEN  PROBLEMS. 

i^— $1.26;   $2.10;   $2.52;   $3.36.     ii— 253 J  bushels. 
13—4:\  da.      U—iL     15— 20-is  da. 

POWERS  AND  ROOTS.  Art.  182.  i— 54.  ^—763.  5—5,453. 
^_.28.  J— .432.  6—U.  7—7.5.  5—60.709.  5— .017.  10— n. 
21—3711  i^— 2.8284+.  i«?— 8.83176+.  i^— 96.5  ft.  i5— 111.75  yd. 
16— dl  in.     i7— 96  ft. ;  336  ft.     i<?— 58  rd.     19—22  rd.  by  88  rd. 

Art.  190.  i— 53.  ^—34.  5— .19.  ^—.87.  5— .08.  6—6.7. 
7—5.63.  <5— 5.04.  5—49.2.  10— i.  11— 2  ^g.  i^— 7.50576+. 
13—5  ft.  8  in.  cube,  i^— 4,916.  15—48  ft.  long,  1  ft.  6'  wide,  and 
3  ft.  high.  i6— 12.9+in.  cube.  17— 6. 135+ in.  cube.  15— 6  ft.  11.42', 
li?— 64. 12+ in.  cube. 

MEASUREMENTS.  Art.  194.  1—145  ft.  7+in.  ;?— 20  ft.  2. 43+ in. 
^_2l .  204  +  feet.  .^—15  feet.  5— 109. 123  +  feet.  6—62  ft.  2. 7  +  in. 
7—25  in.  nearly.  8—9.9  feet.  9—50  feet ;  78  feet ;  136  ft.  8  in. 
10 — 15. 55+ in.  square. 

Art.  201.  1— 5f  sq.  ft.  ;^— 42  sq.  ft.  92  sq.  in.  3—97  sq.  ft.  118f  sq.  in. 
4—201  square  feet  20|  sq.  in.  5—105  sq.  yd.  5  sq.  ft.  139  +  sq.  in. 
6 — 88  square  yards.  7 — 6  sq.  ft.  121J  sq.  in.  5— 124|  square  feet. 
9_5  sq.  ft  127A  sq.  in.  10—82  sq.  yd.  8|  sq.  ft.  11—4  sq.  ft.  63.54  sq.  in. 
1^—4  sq.  ft.  139  sq.  in. 

Art.  202.  1—66  cubic  feet  l,08Hf  cubic  inches.  ^—27  cu.  ft. 
S—S  cu.  ft.  658f  cu.  in.  4— 881. 632+ gal.  5—12  cu.  ft.  l,401f  cu.  in. 
6—66  mi.  945  ^\  yd.  7—6j\  T.  8—1 ,  503  ^V + cu.  in.  £>— 338. 1  +  cu.  in. 
10— 23  tons  1,624  pounds.  11—1,836.734  +  gal. ;  58  bar.  9.734  +  gal. 
1^—256  bar.  14.96  + gal. 

Art.  205.  1—4:  sq.  ft.  131?  sq.  in. ;  1  cu.  ft.  39f  cu.  in.  <^— 12|  sq.  in. 
5— 523.8+cu.  in.  4— 2  sq.  f t.  51  f  sq.  in.  5— 104,991,450, 477^^  cu.  ml 
6—3141  sq.  in. ;  523^  cu.  in.     5—33. 

Art.  206.  1—U  miles.  ;^— 13,068  pounds.  5— 108.86f-  inches. 
^_78.54  sq.  ft.;  113.0976  sq.  in.  5—88  inches.  6—575,000  cu.  ft. 
7_15.118  +  ft.     8—4:  lb.     9—16  in.     16>— 49.152  pailfuls. 


ABBOTTSV  ILLUSTRATED  HISTORIES. 


BIOGRAPHICAL  HISTORIES.  By  Jacob  Abbott  and 
John  S.  C.  Abbott.  The  Volumes  of  this  Series  are  print- 
ed and  bound  uniformly,  and  are  embellished  with  numerous 
Engravings.     16mo,  Cloth,  $1  00  each. 

A  series  of  volumes  containing  severally  full  accounts  of  the  lives  of 
the  most  distinguished  sovereigns,  in  the  various  ages  of  the  world,  from 
the  earliest  periods  to  the  present  day. 

Each  volume  contains  the  life  of  a  single  individual,  and  constitutes  a 
distinct  and  independent  work. 

For  the  convenience  of  buyers  these  popular  Histories  have  been  divid- 
ed into  six  series,  as  follows : 

(Each  series  inclosed  iu  a  neat  Box.    Volumes  may  be  had  separately.) 

4. 
Later  British  Kings  and  Queens. 
Richard  III. 
Mary  Qceen  of  Scots. 
Elizabeth. 
Charles  I. 
Charles  II. 
5. 
Queens  and  Heroines. 
Cleopatra. 
Maria  Antoinette. 
Josephine. 
Hortense. 
Madame  Roland. 

6. 

Rulers  of  Later  Times. 
King  Philip. 
Hernando  Cortez. 
Henry  IV. 
Louis  XIV. 
Joseph  Bonaparte. 
Louis  Philippe. 

Abraham  Lincoln's  Opinion  of  Abbotts'  HisToniES. — In  a  conversation  with 
the  President  just  before  his  death,  Mr.  Lincoln  said :  "  I  want  to  thank  you  and 
your  brother  for  Abbotts'  series  of  Histories.  I  have  not  education  enough  to 
appreciate  the  profound  works  of  voluminous  historians  ;  and  if  I  had,  I  have  no 
time  to  read  them.  But  your  series  of  Histories  gives  me,  in  brief  compass,  just 
that  knowledge  of  past  men  and  events  which  I  need.  I  have  read  them  with 
the  greatest  interest.  To  them  I  am  indebted  for  about  all  the  historical  knowl- 
edge I  have."  

Published  by  HARPER  &  BROTHERS,  New  York. 

tW  Haeper  &  Brothers  will  send  any  of  the  above  works  by  mail,  postage  pre- 
paid,  to  any  part  of  the  United  States,  on  receipt  of  the  price. 


Founders  of  Empires. 
Cyrus. 
Darius. 
Xerxes. 
Alexander. 
Genghis  Khan. 
Peter  the  Great. 

2. 

Hei'oes  of  Roman  History. 
Romulus. 
Hannibal. 
Pyrrhus. 
Julius  C^sar. 
Nero. 
3. 
Earlier  British  Kings  and  Queens. 
Alfred. 

William  the  Conqueror. 
Richard  I. 
Richard  II. 
Margaret  of  Anjou. 


YB  35843 


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